E l e c t ro n ic J o f P r o b a bi l i t y
Electron. J. Probab. 20 (2015), no. 76, 1–47. ISSN: 1083-6489 DOI: 10.1214/EJP.v20-4077
Computing cutoff times of birth and death chains
Guan-Yu Chen
*Laurent Saloff-Coste
†Abstract
Earlier work by Diaconis and Saloff-Coste gives a spectral criterion for a maximum separation cutoff to occur for birth and death chains. Ding, Lubetzky and Peres gave a related criterion for a maximum total variation cutoff to occur in the same setting. Here, we provide complementary results which allow us to compute the cutoff times and windows in a variety of examples.
Keywords: Birth and death chains ; Cutoff phenomenon ; Mixing times. AMS MSC 2010: 60J10 ; 60J27.
Submitted to EJP on January 26, 2015, final version accepted on June 22, 2015. Supersedes arXiv:1502.00361.
1
Introduction
LetX be a finite set andKbe the transition matrix of a discrete time Markov chain onX. Fort ∈ [0, ∞), set Ht= e−t(I−K) = e−t ∞ X i=0 ti i!K i.
If(Xm)∞m=0is a Markov chain onX with transition matrixKandNtis a Poisson process
independent of(Xm)∞m=0with parameter1, thenHt(x, ·)is the distribution ofXNt given
X0= x. It is well-known that ifKis irreducible with stationary distributionπ, then
lim
t→∞Ht(x, y) = π(y), ∀x, y ∈ X .
IfKis assumed further aperiodic, then
lim
m→∞K
m(x, y) = π(y), ∀x, y ∈ X .
For simplicity, we use the triple(X , K, π)to denote a discrete time irreducible Markov chain onX with transition matrixKand stationary distributionπand use(X , Ht, π)to
denote the associated continuous time chain introduced above.
*Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan.
E-mail: [email protected]
†Malott Hall, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, U.S.A.
In this paper, we consider the convergence of Markov chains in both total variation distance and separation. Letµ, νbe two probabilities onX. The total variation distance betweenµ, νand separation ofµw.r.t. νare defined by
kµ − νkTV:= max
A⊂X{µ(A) − ν(A)}, sep(µ, ν) := maxx∈X{1 − µ(x)/ν(x)}.
With initial statex, the total variation distance and separation are defined by
dTV(x, m) := kK
m(x, ·) − πk
TV, dsep(x, m) :=sep(K
m(x, ·), π).
As these quantities are non-increasing inm, it is reasonable to consider the correspond-ing mixcorrespond-ing time, which are defined by
TTV(x, ) := min{m ≥ 0|dTV(x, m) ≤ }
and
Tsep(x, ) := min{m ≥ 0|dsep(x, m) ≤ },
for any ∈ (0, 1). We define the maximum total variation distance and maximum separation by
dTV(m) := max
x∈X dTV(x, m), dsep(m) := maxx∈Xdsep(x, m).
The corresponding mixing times are defined in a similar way and are denoted byTTV()
andTsep(). For the associated continuous time chains, we used
(c) TV ,d (c) sep, T (c) TV andT (c) sep . The inequalities, dTV(m) ≤ dsep(m) ≤ 1 − (1 − 2dTV(m)) 2,
provide comparisons between the maximum total variation distance and maximum separation. As a consequence, one has
TTV() ≤ Tsep() ≤ 2TTV(/4), ∀ ∈ (0, 1).
Those results also apply for the continuous time chain and we refer the reader to [1] for detailed discussions and to [17] for various techniques in estimating the mixing times.
A birth and death chain on{0, 1, ..., n}with transition ratespi, qi, riis a Markov chain
with transition matrixKsatisfying
K(i, i + 1) = pi, K(i, i − 1) = qi, K(i, i) = ri, ∀0 ≤ i ≤ n,
where pi+ qi+ ri = 1andpn = q0 = 0. Conventionally, pi, qi, ri are called the birth,
death and holding rates ati. In the above setting, it is easy to see thatKis irreducible if and only ifpiqi+1> 0for0 ≤ i < nand the unique stationary distributionπsatisfies
π(i) = c(p0· · · pi−1)/(q1· · · qi), wherec is a normalizing constant such thatPiπ(i) = 1.
Ding et al. proved in [14] that, over all initial states, separation is maximized when the chain starts at0ornand Diaconis and Saloff-Coste provided a formula for maximum separation in [12]. As a consequence, the mixing time for maximum separation (and then for the maximum total variation distance) is comparable with the sum of reciprocals of non-zero eigenvalues ofI − K. In [9], Chen and Saloff-Coste showed that both mixing times are of the same order as the maximum expected hitting time to the median ofπ
over all initial distributions concentrated on the boundary points.
The cutoff phenomenon was first observed by Aldous and Diaconis in 1980s. For a formal definition, ifdis the total variation distance or separation either in the maximum case or with a specified initial state, a family of irreducible Markov chains(Xn, Kn, πn)∞n=1
is said to present a cutoff ind, or ad-cutoff, if there is a sequence of positive integers (tn)∞n=1such that ∀ ∈ (0, 1), lim n→∞ Tn,d() tn = 1,
whereTn,dis the mixing time indof thenth chain. A family that presents a cutoff indis
said to have a(tn, bn)cutoff indor a(tn, bn) d-cutoff iftn > 0, bn> 0,bn/tn → 0and
∀ ∈ (0, 1), lim sup
n→∞
|Tn,d() − tn|
bn
< ∞.
In either case, the sequence(tn)∞n=1is called a cutoff time and, in the latter case, the
sequence(bn)∞n=1is called the window with respect to(tn)∞n=1. The definition of cutoffs
for families of continuous time chains is similar and we refer the reader to [11, 6] for an introduction and a detailed discussion of cutoffs. As this article considers the total variation and separation, we refer the reader to [7] for the computation of cutoff times in theL2-distance and to [3] for a refinement of theL2-cutoff locations and window sizes.
Return to birth and death chains. To avoid the confusion of the total variation distances (resp. separation) in the maximum case and with a specified initial states, we useF andFcfor families of birth and death chains without starting states specified and
writeFL, FL c andF
R, FR
c respectively for families of chains started at the left and right
boundary states. Diaconis and Saloff-Coste obtained in [12] a spectral criterion for the existence of the separation cutoff and we cite part of their results in the following.
Theorem 1.1. [12, Theorems 5.1-6.1] Forn = 1, 2, ..., letKn be the transition matrix
of an irreducible birth and death chain on{0, 1, ..., n}andλn,1, ..., λn,nbe the non-zero
eigenvalues ofI − Kn. Set tn = n X i=1 1 λn,i , λn = min 1≤i≤nλn,i, σ 2 n= n X i=1 1 λ2 n,i , ρ2n= n X i=1 1 − λn,i λ2 n,i .
LetF be the family(Kn)∞n=1andFcbe the family of associated continuous time chains.
(1) FL
c has a separation cutoff if and only iftnλn → ∞.
(2) SupposeKn(i, i + 1) + Kn(i + 1, i) ≤ 1for alli, n. Then,FLhas a separation cutoff
if and only iftnλn→ ∞.
Furthermore, iftnλn → ∞, thenFcL has a(tn, σn)separation cutoff and, under the
assumption of (2),FL have a(t
n, max{ρn, 1})separation cutoff.
Remark 1.1. In Theorem 1.1, the(tn, max{ρn, 1})separation cutoff ofFLis not discussed
in [12] but is an implicit result of the techniques therein. We give a proof of this fact in the appendix for completion. In the proof that there is a(tn, max{ρn, 1})separation
cutoff, we show that
FLhas a cutoff ⇔ ρ
n= o(tn) ⇔ max{ρn, 1/λn} = o(tn).
Remark 1.2. For any irreducible birth and death chain, it was proved in [14] that the maximum separation of the associated continuous time chain is attained when the initial state is any of the boundary states. This is also true for the discrete time case if the transition matrixKsatisfiesminiK(i, i) ≥ 1/2. As a result, ifF , Fc andtn, λnare as in
Theorem 1.1, then
(1) Fchas a maximum separation cutoff if and only iftnλn→ ∞.
(2) Assuming thatinfi,nKn(i, i) ≥ 1/2,F has a maximum separation cutoff if and only
For cutoffs in the maximum total variation, Ding, Lubetzky and Peres provide the following criterion in [14].
Theorem 1.2. [14, Corollary 2 and Theorem 3] LetF , Fc, λnbe as in Theorem 1.1 and
letTn,TV, T
(c)
n,TVbe the maximum total variation mixing time of thenth chains.
(1) Fc has a maximum total variation cutoff if and only if T (c)
n,TV()λn → ∞for some
∈ (0, 1).
(2) Assume thatinfi,nKn(i, i) > 0. Then,F has a maximum total variation cutoff if and
only ifTn,TV()λn→ ∞for some ∈ (0, 1).
Remark 1.3. For any birth and death chain, the total variation distance for chain started at the left boundary state can be different from that for chain started at the right boundary state and a biased random walk with constant birth and death rates is a typical example. Further, the maximum total variation distance over all initial states is not necessarily attained at boundary states and a birth and death chain with valley stationary distribution, a distribution which is non-increasing on{0, ..., M } and non-decreasing on{M, ..., n}for some0 < M < n, could illustrate this observation. For instance, let’s consider a birth and death chain on{0, ..., 2n}with transition ratespi = qi = 1/2for
0 < i < 2nandp0= q2n = ∈ (0, 1). It is easy to check that the stationary distributionπ
is given byπ(i) = cfor0 < i < 2nandπ(0) = π(2n) = c/(2)withc = (−1+ 2n − 1)−1. Referring to the notationdTV(x, m)introduced before, it is easy to check that
dTV(0, m) = dTV(2n, m) ≤ dTV(0, 0) = 1 − π(0), ∀m ≥ 0,
and
dTV(n, m) ≥ π({0, 2n}) = 2π(0), ∀0 ≤ m < n.
For < 1/(4n − 2), one has3π(0) > 1, which leads to
dTV(0, m) < dTV(n, m), ∀0 ≤ m < n.
This is very different from the case of separation and we refer the readers to Sections 5 and 6 for more discussions.
To state our main results, we need the following notation. For n ∈ N, let Xn =
{0, 1, ..., n}and(Xm(n))∞m=0be an irreducible birth and death chain onXn with transition
matrixKn and stationary distributionπn. LetNtbe a Poisson process independent of
(Xm(n))with parameter1. Fori ∈ Xn, set
τi(n)= inf{m ≥ 0|Xm(n)= i}, τe
(n)
i = inf{t ≥ 0|X
(n)
Nt = i}. (1.1)
For j ∈ Xn, let Ej and Varj denote the conditional expectation and variance given
X0(n)= j.
Remark 1.4. It follows from the definition of τi(n),eτi(n) that Ejτ (n)
i = Ejτe (n) i for all
i, j ∈ Xn. See [1] for more information of the hitting timesτ (n) i ,τe
(n) i .
Theorem 1.3. LetF , Fc, λnbe as in Theorem 1.1 andτ (n) i ,eτ
(n)
i be the hitting times in
(1.1). Forn ≥ 1, letMn∈ {0, 1, ..., n}and set
sn= E0eτ (n) Mn+ Eneτ (n) Mn= E0τ (n) Mn+ Enτ (n) Mn and b2n =Var0eτ (n) Mn+Varnτe (n) Mn, c 2 n=Var0τ (n) Mn+Varnτ (n) Mn. Suppose that inf n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0. (1.2)
(1) FL
c has a cutoff if and only ifsnλn→ ∞;FcLhas a cutoff if and only ifsn/bn→ ∞.
Furthermore, ifsn/bn → ∞, thenFcLhas a(sn, bn)cutoff.
(2) Assume thatKn(i, i + 1) + Kn(i + 1, i) ≤ 1for all i, n. Then, FL has a cutoff if
and only ifsnλn → ∞;FLhas a cutoff if and only ifsn/cn → ∞. Furthermore, if
sn/cn → ∞, thenFL has a(sn, max{cn, 1/λn})cutoff.
Remark 1.5. Let σn, ρn be the constants in Theorem 1.1. Let Mn, Mn0 ∈ {0, 1, ..., n}
andbn, cn, b0n, c0nbe the constants in Theorem 1.3 defined accordingly. SupposeMn, Mn0
satisfy (1.2). Then,
bn b0n σn, max{cn, 1/λn} max{c0n, 1/λn} max{ρn, 1/λn},
whereun vnmeans that both sequences,un/vnandvn/un, are bounded. See Corollary
2.3 for a proof. Comparing Theorems 1.1 and 1.3, one can see that the cutoff window for
FL
c is unchanged up to some universal multiples but the cutoff window forFLcan have
a bigger order in Theorem 1.3 due to the change of the cutoff time. In total variation, we have the following result.
Theorem 1.4. LetF , Fc, λnbe as in Theorem 1.1 andτ (n) i ,eτ
(n)
i be the hitting times in
(1.1). LetMn∈ {0, 1, ..., n}and set
θn = max n E0τ (n) Mn, Enτ (n) Mn o = maxnE0eτ (n) Mn, Eneτ (n) Mn o and α2n = maxnVar0τe (n) Mn,Varneτ (n) Mn o and β2n= maxnVar0τ (n) Mn,Varnτ (n) Mn o . Suppose inf n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0. (1.3)
In the maximum total variation distance:
(1) Fchas a cutoff if and only ifθnλn→ ∞;Fc has a cutoff if and only ifθn/αn → ∞.
Furthermore, ifFc has a cutoff, thenFchas a(θn, αn)cutoff.
(2) Assume thatinfi,nKn(i, i) > 0. Then,F has a cutoff if and only ifθnλn → ∞;F has
a cutoff if and only ifθn/βn → ∞. Furthermore, ifF has a cutoff, thenF has a
(θn, βn)cutoff.
Remark 1.6. In Theorem 1.4, ifδ = infi,nKn(i, i), thenδα2n≤ β2n≤ α2n. See Remark 5.5
for details.
Remark 1.7. LetF = (Xn, Kn, πn)∞n=1be a family of irreducible birth and death chains
withXn= {0, 1, ..., n}. Fora ∈ (0, 1), setMn(a)be a state inXn satisfying
πn([0, Mn(a)]) ≥ a, πn([Mn(a), n]) ≥ 1 − a.
By Theorem 1.1 and Remark 1.2, ifFchas a cutoff in maximum separation, then
lim n→∞ E0τe (n) Mn(a)+ Eneτ (n) Mn(a) E0eτ (n) Mn(b)+ Eneτ (n) Mn(b) = 1, ∀0 < a < b < 1. (1.4)
From Theorem 1.4, ifFchas a cutoff in the maximum total variation, then
lim n→∞ max{E0eτ (n) Mn(a), Eneτ (n) Mn(a)} max{E0τe (n) Mn(b), Eneτ (n) Mn(b)} = 1, ∀0 < a < b < 1. (1.5)
But, the converse of these statements are not necessarily true. For example, let
Kn(i, i + 1) = Kn(i + 1, i) = 1/2, ∀0 < i < n, Kn(n, n) = 1/2,
and
Kn(0, 1) = Kn(1, 0) = ξn, Kn(0, 0) = 1 − ξn, Kn(1, 1) = 1/2 − ξn,
whereξn ∈ (0, 1/2). Note thatKncan be regarded as the transition matrix of a simple
random walk onXn with specific transitions at the boundary states and a bottleneck
between0and1whenξn is small. It is clear that the stationary distribution satisfies
πn(i) = 1/(n + 1)for all0 ≤ i ≤ n. After some computations, one has, fornlarge enough,
Mn(a) n (n − Mn(a)). This implies E0τe (n) Mn(a)= 1 ξn + Mn(a)(Mn(a) + 1) − 2 1 ξn + n2 and Eneτ (n)
Mn(a)= (n − Mn(a))[n − Mn(a) + 1] n
2.
Letpn,i, qn,i, rn,i andλnbe the transition rates and the spectral gap ofKn. By Theorem
1.2 in [9], we have 1 λn max max j:j<Mn Mn−1 X k=j πn([0, j]) πn(k)pn,k , max j:j>Mn j X k=Mn+1 πn([j, n]) πn(k)qn,k , (1.6)
whereMn= bn/2c. This implies
1 λn
1
ξn
+ n2.
As a consequence of Theorems 1.3 and 1.4,Fchas neither a maximum separation cutoff
nor a maximum total variation cutoff. Letsn andθn be the constants in Theorems 1.3
and 1.4. Ifn2ξn→ 0, then sn∼ θn∼ E0τe (n) Mn(a)∼ 1 ξn , ∀a ∈ (0, 1).
The above example illustrates that (1.4) and (1.5) are necessary but not sufficient for the existence of the corresponding cutoffs.
One can see from Theorems 1.3 and 1.4 that, in general, the cutoff phenomenon occurs when the first hitting times to some large sets are concentrated on their expected values. We refer the reader to [4] for more general results in similar heuristics and to [16] for some other relationship between the cutoffs and the hitting times.
The following theorem describes one of the main applications of Theorems 1.3-1.4.
Theorem 1.5. Consider a familyF = (Xn, Kn, πn)∞n=1 of irreducible birth and death
chains with Xn = {0, 1, ..., n}. For n ≥ 1, let (Ωn, P(n)) be a probability space and
Cn,1, ..., Cn,n: Ωn→ (0, 1)be independent and identically distributed random variables.
Forωn∈ Ωnand0 ≤ i ≤ n, let(Xn, L (ωn)
n , πn)be a Markov chain given by
L(ωn)
n (i, i + 1) = Kn(i, i + 1)Cn,i+1(ωn),
L(ωn) n (i + 1, i) = Kn(i + 1, i)Cn,i+1(ωn), L(ωn) n (i, i) = 1 − L (ωn) n (i, i + 1) − L (ωn) n (i, i − 1),
and, for ω = (ω1, ω2, ...) ∈ Qn=1∞ Ωn, let F(ω) = (Xn, L (ωn)
n , πn)∞n=1. Let Fc, F (ω) c be
the continuous time families associated withF , F(ω). For n ≥ 1, setµ
n = E(1/Cn,1),
(1) IfFc has a maximum total variation cutoff andνnαn = o(µnθn), then there is a
sequenceEn ⊂ Ωnsuch thatP(n)(En) → 1and, for anyω ∈Q ∞
n=1En,F (ω) c has a
maximum total variation cutoff with cutoff timeµnθn.
(2) Assuminginfn,iKn(i, i) > 0and replacingαnbyβn, the statement in (1) also holds
for the familiesF , F(ω).
Remark 1.8. In Theorem 1.5,Ln can be regarded as a random birth and death chain
obtained by applying i.i.d. random slowdowns onKnwithout changing the stationary
distribution.
Remark 1.9. Theorem 1.5 also holds in maximum separation.
The remaining of this article is organized in the following way. Sections 2 and 3 contain the proofs of Theorems 1.3 and 1.4 respectively. The proof of Theorem 1.5 is given in Section 4. We also introduce another randomization of simple random walks on paths and discuss its cutoff and mixing time. In Section 5, we consider families of chains started at one boundary states and provide criteria for the existence of a total variation cutoff and formulas for the cutoff time. We discuss the distinction between maximum total variation cutoffs and cutoffs from a boundary state and illustrate this with several examples in Section 6. The main results of Section 5 are proved in Section 7. In Section 8, we apply the developed theory to compute the cutoff time of some classical examples. As some of the illustrated examples might be interesting to some readers, we would like to highlight this section, though it is placed after those long proofs in Section 7. Some useful lemmas and auxiliary results are gathered in the appendix.
2
Cutoff in separation
This section is dedicated to the proof of Theorem 1.3 and we need the following two lemmas. The first lemma concerns the mean and variance of hitting times and the second lemma provides a comparison of spectral gaps.
Lemma 2.1. LetKbe the transition matrix of an irreducible birth and death chain on
{0, 1, ..., n}. For1 ≤ i ≤ n, letβ1(i), ..., βi(i)be the eigenvalues of the submatrix ofI − K
indexed by{0, ..., i − 1}and set
τi = min{m ≥ 0|Xm= i}, eτi= inf{t ≥ 0|XNt= i}, (2.1)
where(Xm)∞m=0is a Markov chain with transition matrixKandNtis a Poisson process
independent ofXmwith parameter1. Then,β (i)
j ∈ (0, 2)for all1 ≤ j ≤ iand
E0τi= E0τei= i X j=1 1 βj(i) , (2.2) and Var0(τi) = i X j=1 1 − βj(i) β(i)j 2, Var0(eτi) = i X j=1 1 βj(i) 2. (2.3)
Proof. LetKe be the submatrix ofKindexed by{0, 1, ..., i − 1}. Letβbe an eigenvalue of e
Kandx = (x0, ..., xi−1)be a left eigenvector associated withβ. That is,
βxj = K(j − 1, j)xj−1+ K(j, j)xj+ K(j + 1, j)xj+1, ∀0 < j < i − 1, βx0= K(0, 0)x0+ K(1, 0)x1,
By the irreducibility ofK, ifxi−1= 0, thenxj= 0for all0 ≤ j < i. This impliesxi−16= 0 and then |β| i−1 X j=0 |xj| ≤ i−1 X j=0
|xj| − K(i − 1, i)|xi−1| < i−1
X
j=0
|xj|.
Sincexis an eigenvector ofK,P
j|xj| > 0and thus|β| < 1. This proves thatβ (i)
j ∈ (0, 2)
for all1 ≤ j ≤ i. For (2.2) and (2.3), note that the distribution ofeτi was given by Brown
and Shao in [5] and the technique therein also applies forτi. This leads to the desired
identities, where we refer the reader to their work for details. Remark 2.1. In Lemma 2.1, the first equality of (2.3) implies
i X j=1 1 (βj(i))2 ≥ i X j=1 1 βj(i) , ∀j ≥ 1.
Lemma 2.2. LetKbe the transition matrix of an irreducible birth and death chain on
{0, 1, ..., n}with stationary distributionπ. For0 ≤ i ≤ n, letLibe the sub-matrix ofK
obtained by removing the row and column ofK indexed by statei. Letλ1 < · · · < λn
be the non-zero eigenvalues ofI − Kandλ(i)1 ≤ · · · ≤ λ(i)n be the eigenvalues ofI − Li.
Then,
λ(i)j ≤ λj ≤ λ (i)
j+1≤ λj+1, ∀1 ≤ j < n,
and
min{π([0, i]), π([i, n])} 4
λ1≤ λ
(i) 1 ≤ λ1.
In particular, if M is a median of π, i.e. π([0, M ]) ≥ 1/2 and π([M, n]) ≥ 1/2, then
λ1/8 ≤ λ (M ) 1 ≤ λ1.
The proof of Lemma 2.2 is based on a weighted Hardy inequality obtained in [9] and is discussed in the appendix. In what follows, for any two sequences of positive reals
an, bn, we writean= o(bn)ifan/bn→ 0and writean= O(bn)ifan/bn is bounded. In the
case thatan= O(bn)andbn = O(an), we writean bninstead.
Proof of Theorem 1.3. Letλn,i, λn, tn, σn, ρnbe constants in Theorem 1.1. Note that, for
n ≥ 2, max{ρ2n, 1/λ2n} ≤ σ2 n= n X i=1 1 λ2n,i ≤ tn λn . This implies p tnλn ≤ tn σn ≤ tn max{ρn, 1/λn} ≤ tnλn. (2.4) As a consequence, we have
tnλn→ ∞ ⇔ σn= o(tn) ⇔ max{ρn, 1/λn} = o(tn). (2.5)
Next, letsn, bn, cnbe constants in Theorem 1.3. Observe that
1/λn ≤ max{ρn, 1/λn} ≤ σn.
Setan= min{πn([0, Mn]), πn([Mn, n])}. By Lemmas 2.1 and 2.2, one has
tn≤ sn≤ tn+ 4 anλn ≤ tn+ 4σn an
and σ2n≤ b2 n ≤ σ 2 n+ 4 anλn 2 ≤17σ 2 n a2 n .
According to the assumption of (1.2), we havean 1and this implies
tnλn → ∞ ⇔ snλn→ ∞
and
|tn− sn| = O(σn), |tn− sn| = O(max{ρn, 1/λn}), bn σn. (2.6)
As a consequence of (2.5) and (2.6), we obtain
tnλn→ ∞ ⇔ bn= o(sn) ⇔ max{cn, 1/λn} = o(sn). (2.7)
The first equivalence of (2.7) proves the criterion for cutoff in (1). For (2), ifFL has a
separation cutoff, then Theorem 1.1 impliestnλn → ∞. By the last identity in (2.7), we
obtaincn= o(sn). To see the inverse direction, observe that the mappingu 7→ (1 − u)/u2
is decreasing on(0, 2]andλn,i∈ (0, 2)for all1 ≤ i ≤ n. In the same reasoning as before,
Lemmas 2.1 and 2.2 yield
ρ2n ≤ c2 n≤ ρ 2 n+ 1 − anλn/4 (anλn/4)2 +λn,n− 1 λ2 n,n ≤ ρ2 n+ 17 a2 nλ2n . (2.8)
By the first inequality of (2.8), ifcn = o(sn), thenρn = o(sn). Accompanied with the facts,
sn = tn+ 4 anλn ≤ 1 + 4 an tn, an 1,
we obtainρn= o(tn). By Remark 1.1,FLhas a separation cutoff.
To see a window, we recall Corollary 2.5(v) of [6], which says that if a family has a
(tn, σn)cutoff and
bn = o(tn) (orbn= o(sn)), |tn− sn| = O(bn), σn= O(bn),
then the family has a(sn, bn)cutoff. By Theorem 1.1, the desired cutoff forFcLis given
by the first and third identities in (2.6), while the desired cutoff forFLis provided by
the second identity in (2.6), the third identity in (2.7) and the following observations
max{ρn, 1/λn} max{cn, 1/λn}, max{ρn, 1} = O(max{cn, 1/λn}),
which are implied by (2.8) and the factλn ≤ 2.
In the following corollary, we summarize some useful comparison between the vari-ances of hitting times and the windows of cutoffs obtained in the proof of Theorem 1.3.
Corollary 2.3. LetKbe the transition matrix of an irreducible birth and death chain on
{0, 1, ..., n}with stationary distributionπandτi,τeibe the hitting times in (2.1). Suppose λ1, ..., λnbe non-zero eigenvalues ofI − Kand set
t = n X i=1 1 λi , σ2= n X i=1 1 λ2i, ρ 2= σ2− t, λ = min 1≤i≤nλi. Then, for0 ≤ i ≤ n, t ≤ E0τei+ Enτei= E0τi+ Enτi≤ t + 4 a(i)λ and σ2≤Var0eτi+Varneτi≤ 17σ2 a(i)2, ρ 2≤Var 0τi+Varnτi ≤ ρ2+ 17 a(i)2λ2,
To determine a cutoff time and a window using Theorem 1.3, one needs to compute the mean and variance of the hitting time to some state given that the chain starts at one boundary state. Explicit formulas on both terms are available using the Markov property and we summarize them in Lemma A.1.
The next proposition discusses the cutoff times obtained in Theorem 1.3 and provides a universal lower bound on the corresponding windows using the transition rates and the stationary distribution.
Proposition 2.4. LetKbe the transition matrix of a birth and death chain on{0, 1, ..., n}
with transition ratespi, qi, ri. Letτi,eτi be the hitting times in (2.1) and set s(i) = E0τei+ Enτei, b(i)2=Var0(eτi) +Varn(eτi).
SupposeK is irreducible with stationary distributionπand spectral gapλ. LetM ∈ {0, 1, ..., n}be a state satisfyingπ([0, M ]) ≥ 1/2 andπ([M, n]) ≥ 1/2. Then, for0 ≤ i ≤ j ≤ M, s(i) − s(j) = j−1 X `=i 1 − 2π([0, `]) p`π(`) ≥ 0, (2.9) and, for0 ≤ i ≤ n, b(i) ≥ 1 λ≥ 1 20≤j≤M ≤k≤nmax max M −1 X `=j π([0, j]) p`π(`) , k X `=M +1 π([k, n]) q`π(`) . (2.10)
Proof. (2.9) is given by Lemma A.1 and the first inequality of (2.10) is obvious from Lemmas 2.1-2.2, while the second inequality of (2.10) is cited from Theorem A.1 of [9].
Remark 2.2. Letsn, tn be the constants in Theorems 1.1-1.3. By Corollary 2.3, one has
sn− tn≥ 0and, by (2.9), the differencesn− tnis minimized whenMn satisfies
πn([0, Mn]) ≥ 1/2, πn([Mn, n]) ≥ 1/2.
3
Cutoff in total variation
This section is dedicated to the proof of Theorem 1.4. Throughout the rest of this article, we will writePito denote the probability given the initial statei. First, recall
two useful bounds on the total variation.
Lemma 3.1. [9, Proposition 3.8 and Equation (3.5)] Consider a continuous time birth
and death chain on{0, 1, ..., n}with stationary distributionπ. For 0 ≤ i ≤ n, leteτi be
the first hitting time to stateiandd(c)TV (i, t)be the total variation distance at timetwith
initial statei. Then, for0 ≤ i ≤ nand0 ≤ j ≤ k ≤ n,
d(c)TV (i, t) ≤ Pi(max{eτj,τek} > t) + 1 − π([j, k])
and
d(c)TV (0, t) ≥ P0(eτi> t) − π([0, i − 1]).
Based on the above lemma, we may bound the maximum total variation mixing time using the expected hitting times.
Theorem 3.2. Letπ,eτibe as in Lemma 3.1 and set
θ(i) = max{E0eτi, Eneτi}, α(i)
2= max{Var
The maximum total variation mixing time satisfies, for any0 ≤ j ≤ k ≤ nandδ ∈ (0, 1), TTV(c)(1) ≤ θ(j) + Ejeτk+ Ekτej+ q 2 δ − 1 max{α(j), α(k)} and TTV(c)(2) ≥ θ(j) − Ekeτj− q 1 δ − 1 max{α(j), α(k)},
where1= 1 − π([j, k]) + δand2= min{π([j, n]), π([0, k])} − δ.
Proof. We first consider the upper bound. Set1 = 1 − π([j, k]) + δ. By Lemma 3.1, if
i ≤ j, then
d(c)TV (i, t) ≤ P0(eτk > t) + 1 − π([j, k]).
As a result of the one-sided Chebyshev inequality, this implies
TTV(c)(i, 1) ≤ E0eτk+ q 1 δ − 1α(k). Similarly, ifi ≥ k, then TTV(c)(i, 1) ≤ Enτej+ q 1 δ − 1α(j).
Note that, in the casej < i < k,
Pi(max{τej,τek} > t) ≤ Pi(eτk> t) + Pi(eτj> t) ≤ Pj(τek> t) + Pk(eτj > t). This implies TTV(c)(i, 1) ≤ Ejeτk+ Ekτej+ q 2 δ − 1 max{α(j), α(k)}.
Combining all above gives the desired upper bound.
For the lower bound, set2= min{π([j, n]), π([0, k])} − δ. By the second inequality of
Lemma 3.1, one has
d(c)TV (0, t) ≥ π([j, n]) − P0(eτj ≤ t).
Settingt = E0eτj−p(1/δ − 1)α(j)in the above inequality derives
d(c)TV (0, t) ≥ π([j, n]) − δ ≥ 2. This implies TTV(c)(2) ≥ T (c) TV (0, 2) ≥ E0τej− q (1δ − 1)α(j).
Similarly, fork ≥ j, we have
TTV(c)(2) ≥ Eneτk− q
(1δ − 1)α(k) = Eneτj− Ekeτj− q
(1δ − 1)α(k).
Both inequalities combine to the desired lower bound.
Proof of Theorem 1.4(Continuous time case). It has been shown in [14] that separation is maximized when the chain started at any of the boundary states and the maximum total variation cutoff is equivalent to the maximum separation cutoff. It is clear that the constants,snandbn, in Theorem 1.3 are respectively of the same order as the constants,
θn andαn, in Theorem 1.4. As a consequence of Theorem 1.3,Fc has a cutoff in the
maximum total variation if and only ifθnλn→ ∞if and only ifθn/αn → ∞.
To see a cutoff time and a window, we assume in the following thatθn/αn→ ∞. Set
0= inf
For ∈ (0, 0), we may choosexn, yn such that πn([0, xn]) ≥ 3, πn([xn, n]) ≥ 1 − 3, πn([0, yn]) ≥ 1 − 3, πn([yn, n]) ≥ 3.
Clearly,xn≤ yn. Replacingj, k, δwithxn, yn, /3in Theorem 3.2 yields
Tn,(c)TV() ≤ θn(xn) + Exneτ (n) yn + Eyneτ (n) xn + r 6 max{αn(xn), αn(yn)}, where θn(j) := max{E0eτ (n) j , Eneτ (n) j }, α 2 n(j) = max{Var0τe (n) j ,Varneτ (n) j }.
In the above notations,θn= θn(Mn)andαn= αn(Mn). Sincexn≤ Mn≤ yn, one has
Eneτ (n) xn = Enτe (n) Mn+ EMnτe (n) xn, E0eτ (n) Mn= E0τe (n) xn + Exnτe (n) Mn.
Note that, for any positive realsa, b, c, d,
| max{a + b, c} − max{a, c + d}| ≤ max{b, d}.
This implies |θn(xn) − θn| ≤ Exnτe (n) Mn+ EMneτ (n) xn ≤ Exneτ (n) yn + Eyneτ (n) xn.
According to the definition ofxn, yn, Mn, Corollary 2.3 implies
αn(xn) αn αn(yn).
Letpn,`, qn,`be the birth and death rates of thenth chain. The replacement ofj, M, k
withxn, Mn, yn in (2.10) yields that, for any0 ≤ i ≤ n,
αn(i) ≥ 1 2√2max (Mn−1 X `=xn πn([0, xn]) pn,`πn(`) , yn X `=Mn+1 πn([yn, n]) qn,`πn(`) ) ≥ 12√2 yn−1 X `=xn 1 pn,`πn(`) = 12√2 yn X `=xn+1 1 qn,`πn(`) ≥ 12√2max{Exneτ (n) yn , Eynτe (n) xn},
where the second inequality uses the factqn,`πn(`) = pn,`−1π(`−1)and the last inequality
applies the first identity in Lemma A.1. As a consequence, we may conclude from the above discussions that
Tn,(c)TV() − θn ≤ 48√2 + r 6 ! max{αn(xn), αn(yn)} αn,
for all ∈ (0, 0). In a similar statement, one can show, by the second part of Theorem
3.2, that θn− T (c) n,TV(1 − ) ≤ 36√2 + r 3 ! max{αn(xn), αn(yn)} = O(αn),
Proof of Theorem 1.4(Discrete time case). We will use the result in the continuous time case and [8] to deal with the discrete time case. Set
δ = inf
n,iKn(i, i), K (δ)
n = (Kn− δI)/(1 − δ).
In the assumption for discrete time case, we have δ ∈ (0, 1). Let Xn = {0, 1, ..., n},
F(δ) = (X n, K
(δ)
n , πn)∞n=1 andF (δ)
c be the family of continuous time chains associated
withF(δ). It was proved in [8] (See Theorems 3.1 and 3.3) that, in the maximum total
variation,
F has a cutoff ⇔ Fc(δ)has a cutoff (3.1)
and
F has a(tn, bn)cutoff ⇔ Fc(δ) has a((1 − δ)tn, bn)cutoff. (3.2)
Leteτi(n,δ)be the hitting time to stateiof the continuous time chain associated withKn(δ)
andEi,Varibe the conditional expectation and variance given the initial statei. Set
θ(δ)n = maxnE0τe (n,δ) Mn , Eneτ (n,δ) Mn o , βn(δ)= maxnVar0eτ (n,δ) Mn ,Varnτe (n,δ) Mn o .
ForFc(δ), it has been proved in the continuous time case that
F(δ) c has a cutoff ⇔ θ (δ) n λ (δ) n → ∞ ⇔ θ (δ) n /β (δ) n → ∞,
whereλ(δ)n is the smallest non-zero eigenvalue ofI − Kn(δ). Furthermore, if it holds true
thatθn(δ)/βn(δ)→ ∞, thenFc(δ)has a(θ(δ)n , βn(δ))cutoff. As a result of (3.1) and (3.2), we
have
F has a cutoff ⇔ θ(δ)n /βn(δ)→ ∞,
and, further, if the right side holds, thenFhas a(θn(δ)/(1 − δ), βn(δ))cutoff.
Letλn, θn, βn be the constants in Theorem 1.4. Clearly,λn = (1 − δ)λ (δ)
n . To finish the
proof, it suffices to show that
θ(δ)n = (1 − δ)θn, β(δ)n βn. (3.3)
Letpn,i, qn,i, rn,ibe the transition rates ofKnandp (δ) n,i, q
(δ) n,i, r
(δ)
n,i be the transition rates of
Kn(δ). It is clear that p(δ)n,i= pn,i/(1 − δ), q (δ) n,i = qn,i/(1 − δ), r (δ) n,i = (rn,i− δ)/(1 − δ).
The first equality of (3.3) is an immediate result of the first identity of Lemma A.1. To see the second part of (3.3), letλn,1, ..., λn,nbe eigenvalues of the submatrix ofI − Kn
obtained by removing theMn-th row and column. Clearly,λn,1/(1 − δ), ..., λn,n/(1 − δ)
are eigenvalues of the submatrix ofI − Kn(δ)obtained by removing theMn-th row and
column. As a consequence of Lemma 2.1, we have
β2n n X i=1 1 − λn,i λ2 n,i , βn(δ) 2 n X i=1 1 λ2 n,i .
Note that the application of Remark 2.1 on the chain(Xn, K (δ) n , πn)says (1 − δ) n X i=1 1 λ2 n,i ≥ n X i=1 1 λn,i . This impliesβn β (δ) n .
4
A randomization of birth and death chains
This section gives two nontrivial examples as applications of theorems in the intro-duction. The first example is stated in Theorem 1.5 and we discuss its proof in the following.
Proof of Theorem 1.5. The proofs for Fc andF are similar and we consider only the
continuous time case. Let Mn, θn, αn be as in Theorem 1.4. For convenience, we let
(pn,i, qn,i, rn,i)be the transition rates ofKn. Forn ≥ 1, set
θn,1= Mn−1 X i=0 πn([0, i]) πn(i)pn,i , θn,2= n X i=Mn+1 πn([i, n]) πn(i)qn,i and α2n,1= Mn−1 X i=0 Mn−1 X j=i πn([0, i])2 πn(i)pn,iπn(j)pn,j , αn,22 = n X i=Mn+1 i X j=Mn+1 πn([i, n])2 πn(i)qn,iπn(j)qn,j .
It is clear from Lemma A.1 that
θn= max{θn,1, θn,2}, αn= max{αn,1, αn,2}.
Without loss of generality, we may assume thatθn = θn,1. Forn ≥ 1, letUn,1, Vn,1 be
positive random variables defined by
Un,1= Mn−1
X
i=0
πn([0, i])
πn(i)pn,iCn,i+1
, Vn,12 = Mn−1 X i=0 Mn−1 X j=i πn([0, i])2
πn(i)pn,iCn,i+1πn(j)pn,jCn,j+1
.
By the independency ofCn,i, one may compute
EUn,1= µnθn,1= µnθn, Var(Un,1) = ν2nα 2 n,1≤ ν 2 nα 2 n and EVn,12 = X 0≤i<j≤Mn−1 πn([0, i])2 πn(i)pn,iπn(j)pn,j µ2n+ Mn−1 X i=0 πn([0, i])2 πn(i)2p2n,i (µ2n+ νn2) ≤ (µ2n+ ν 2 n)α 2 n,1≤ [(µn+ νn)αn,1]2.
The above estimation ofEVn,12 implies
EVn,1≤ q EV2 n,1≤ (µn+ νn)αn,1≤ (µn+ νn)αn. Setan=p(µnθn)/(νnαn),bn=p(µnθn)/[(µn+ νn)αn]and En,1= {ωn ∈ Ωn: |Un,1(ωn) − µnθn| < anνnαn, Vn,1(ωn) < bn(µn+ νn)αn}.
SinceFchas a maximum total variation cutoff, Theorem 1.4 impliesαn= o(θn). In the
assumption of(νnαn) = o(µnθn), it is easy to see that, forωn ∈ En,1,
Un,1(ωn) ∼ µnθn, Vn,1(ωn) = o(µnθn).
By the Chebyshev and Markov inequalities, the fact thatan, bn→ ∞yieldsP(n)(En,1) →
In the same way, we set Un,2= n X i=Mn+1 πn([i, n])
πn(i)qn,iCn,i
, Vn,22 = n X i=Mn+1 i X j=Mn+1 πn([i, n])2
πn(i)qn,iCn,iπn(j)qn,jCn,j
,
and
En,2= {ωn ∈ Ωn: Un,2(ωn) < µnθn+ anνnαn, Vn,2(ωn) < bn(µn+ νn)αn}.
A similar reasoning as before yields thatP(n)(E
n,2) → 1and, forωn ∈ En,2,
Un,2(ωn) ≤ µnθn(1 + o(1)), Vn,2(ωn) = o(µnθn).
As consequence, if we setEn= En,1∩ En,2, thenP(n)(En) → ∞and, forωn ∈ En,
max{Un,1, Un,2} ∼ µnθn, max{Vn,1, Vn,2} = o(µnθn).
The maximum total variation cutoff forFc(ω)and the cutoff timeµnθnare immediate from
Theorem 1.4.
Remark 4.1. From the proof given above, one can derive a variation of Theorem 1.5. Namely, under the assumption ofνnαn= o(µnθn), ifFchas no maximum total variation
cutoff (resp. maximum separation cutoff), then there is a sequenceEn ⊂ Ωn satisfying
P(n)(E
n) → 1 such thatF
(ω)
c has no maximum total variation cutoff (resp. maximum
separation cutoff) forω ∈ Q∞
n=1En. Note that, the requirementνnαn = o(µnθn)and
the assumption of no cutoff will imply the existence of a subsequence, sayin, such that
νin = o(µin). As a result of the Chebyshev inequality,1/Cin,1− E(1/Cin,1)converges in
probability to0. This turnsFc(ω)into a lazy version ofFcwith high probability.
Note that the hypothesis of νnαn = o(µnθn) requires the existence of a second
moment of1/Cn,1. Next, we give an example where1/Cn,1 does not have a finite first
moment.
Theorem 4.1. Forn ≥ 1, letCn,1, ..., Cn,nbe i.i.d. uniform random variables over(0, 1)
defined on(Ωn, P(n)). Forω = (ω1, ω2, ...) ∈ QnΩn, letF(ω) = (Xn, K (ωn)
n , πn)∞n=1be a
family of birth and death chains withXn= {0, 1, ..., n}and
( K(ωn)
n (i, i + 1) = K(i + 1, i) = Cn,i+1/2, ∀0 ≤ i < n,
K(ωn)
n (i, i) = 1 − Kn(ωn)(i, i + 1) − Kn(ωn)(i, i − 1), ∀i.
LetFc(ω)be the family of continuous time chains associated withF(ω)and, forωn∈ Ωn,
let Tc
n,TV(ωn, ·)be the maximum total variation mixing time for (Xn, K
(ωn)
n , πn). Then,
there is a sequenceEn ⊂ Ωn satisfyingP(n)(En) → 1such that, for anyω = (ω1, ω2, ...) ∈
Q∞
n=1En, the familyF (ω)
c has no maximum total variation cutoff andTn,cTV(ωn, ) n
2log n
for ∈ (0, 1/10).
Proof. LetMn ∈ Xn andUn,1, Un,2be as in the proof of Theorem 1.5. Forn ≥ 1, set
Ωn= Cn,i> 1 n log n, ∀1 ≤ i ≤ n , P(n)(·) = P(n)(·|Ωn),
where P(n) is the conditional probability ofP(n) givenΩn. Clearly, P(n)(Ωn) = (1 −
1/n log n)n → 1 and, in
P(n), Cn,1, ..., Cn,n are i.i.d. random variables uniformly
dis-tributed over(1/n log n, 1). LetEand Var be the expectation and variance taken inP(n). It is an easy exercise to compute
E(1/Cn,1) =
log n + log log n
and
Var(1/Cn,1) = n log n − (E(1/Cn,1))2∼ n log n,
This implies that, ifMn→ ∞andn − Mn→ ∞, then
EUn,1∼ Mn2log n, EUn,2∼ (n − Mn)2log n,
and
Var(Un,1) ∼ Mn2n log n, Var(Un,2) ∼ (n − Mn)2n log n.
Fora ∈ (0, 1), ifMn= banc, we writeU (a)
n,i forUn,i. As a result of the above computation,
we obtain EUn,1(a)∼ a 2n2log n, EUn,2(a)∼ (1 − a) 2n2log n, and
Var(Un,1(a)) ∼ a2n3log n, Var(Un,2(a)) ∼ (1 − a)2n3log n.
Forn ≥ 1, let En= n ωn∈ An: |U (a) n,1− a
2n2log n| < n3/2log n, fora = 1/4, 1/2o.
It is easy to show that P(n)(En) → 1 and, hence, P(n)(En) ≥ P(n)(An)P (n) (En) → 1. Furthermore, forωn∈ En, max{Un,1(1/2)(ωn), U (1/2) n,2 (ωn)} ∼ n2log n 4 and max{Un,1(1/4)(ωn), U (1/4) n,2 (ωn)} ∼ 9n2log n 16 .
By Remark 1.7,Fc(ω)has no maximum total variation cutoff forω ∈QnEn. The order of
the mixing time is given by Theorems 3.1 and 3.9 of [9].
Remark 4.2. We refer the reader to [13, 19, 20] for other randomized birth and death chains, which are different from the one considered in Theorem 4.1.
5
Chains started at boundary states
For continuous time birth and death chains, [14] shows that separation reaches its maximum when the initial state is any of the boundary states. This is not true in the case of total variation and it is easy to construct counterexamples. In this section, we discuss the total variation cutoff for families of birth and death chains started at a boundary state. As before, we useFandFc for families of birth and death chains without starting
states specified and writeFL, FL
c andFR, FcRrespectively for families of chains started
at the left and right boundary states.
The following theorem displays a list of equivalent conditions for the total variation cutoff. It is worthwhile to note that some of these conditions are very similar to the conditions in Theorem 1.4.
Theorem 5.1. LetF = (Xn, Kn, πn)∞n=1be a family of irreducible birth and death chains
withXn= {0, 1, ..., n}andFcbe the family of associated continuous time chains inF. For
n ≥ 1, leteτi(n)be the first hitting time to stateiof thenth chain inFcand, fora ∈ (0, 1),
letMn(a)be a state inXn satisfying
and letλn(a)be the smallest eigenvalue of the submatrix ofI − Knindexed by states 0, ..., Mn(a) − 1. Set un(a) = E0τe (n) Mn(a), v 2 n(a) =Var0eτ (n) Mn(a).
Assume thatπn(0) → 0. Then, the following are equivalent.
(1) FL
c has a total variation cutoff.
(2) un(a)/vn(a) → ∞for alla ∈ (0, 1).
(3) un(a)λn(a) → ∞for alla ∈ (0, 1).
(4) There area ∈ (0, 1)and a positive sequence(tn)∞n=1satisfying
tn= O(un(c)), ∀c ∈ (0, 1) and lim n→∞P0 e τM(n) n(a)> (1 − )tn = 1, ∀ ∈ (0, 1),
and, for anyb ∈ (a, 1), there isαb∈ (0, 1)such that
lim sup n→∞ P0 e τM(n) n(b)> (1 + )tn ≤ αb, ∀ > 0,
wherePidenotes the probability given the initial statei.
Furthermore, if (2) or (3) holds, thenFL
c has a cutoff with cutoff time(un(a))∞n=1for
anya ∈ (0, 1). If (4) holds, thenFL
c has a cutoff with cutoff time(tn)∞n=1.
The discrete time version of the previous theorem can be stated as follows.
Theorem 5.2. LetF , Mn(a), λn(a)be as in Theorem 5.1. Forn ≥ 1, letτ (n)
i be the first
hitting time to stateiof thenth chain inFand, fora ∈ (0, 1), set
un(a) = E0τ (n) Mn(a), w 2 n(a) =Var0τ (n) Mn(a).
Assume thatπn(0) → 0,infi,nKn(i, i) > 0andun(a) → ∞for somea ∈ (0, 1). Then, the
conclusion in Theorem 5.1 remains true for the familyFL with the replacement ofv n(a)
bywn(a).
Remark 5.1. The proofs of Theorems 5.1 and 5.2 are complicated and are given in Section 7. It is shown in the beginning of those proofs that the conditionπn(0) → 0is
necessary for the existence of cutoff ofFL
c andFL.
Remark 5.2. LetF , Fc be as in Theorem 5.1 and(pn,i, qn,i, rn,i)be the transition rates of
thenth chains inF. LetMn∈ Xnbe a sequence of states satisfying (1.3), that is,
inf
n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0,
andxn∈ {0, n}be a boundary state fulfilling the following equation
max{E0τe (n) Mn, Enτe (n) Mn} = Exnτe (n) Mn.
By Lemma A.1 and Theorem A.1 of [9], ifxn= 0, then
Exneτ (n) Mn = Mn−1 X i=0 πn([0, i]) πn(i)pn,i ≤ Mn−1 X i=0 1 πn(i)pn,i
and 1 λn ≥ min{πn([0, Mn]), πn([Mn, n])} × max j:j<Mn Mn−1 X i=j πn([0, j]) πn(i)pn,i ≥ min{πn([0, Mn]), πn([Mn, n])} × πn(0) Mn−1 X i=0 1 πn(i)pn,i This implies Exneτ (n) Mnλn≤ 1 min{πn([0, Mn]), πn([Mn, n])}πn(0) .
In a similar way, this inequality also holds in the casexn = n. As a consequence of
Theorem 1.4, ifFc has a maximum total variation cutoff, thenπn(xn) → 0. The above
discussion also holds forF with the assumptioninfn,iKn(i, i) > 0.
Remark 5.3. LetFL
c andFLbe the families in Theorems 5.1 and 5.2. IfFcL(resp. FL)
has a total variation cutoff with cutoff timetn(resp. tn → ∞), then
tn∼ E0eτ
(n)
Mn, (resp. tn ∼ E0τ
(n) Mn, )
whereMn∈ Xnis any sequence satisfying
inf
n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0. (5.1)
In particular, ifFL
c (resp.FL) has a total variation cutoff with bounded cutoff time, then
one may use Lemma 3.1 and Theorem 5.1 to derive
E0eτ (n)
Mn= O(1), (resp. E0τ
(n)
Mn= O(1), )
for any sequenceMn∈ Xn satisfying (5.1).
Remark 5.4. LetFL
c be the family in Theorems 5.1. IfF L
c has a total variation cutoff,
thenun(a) ∼ un(b)for alla, b ∈ (0, 1), or equivalently
EMn(a)eτ (n) Mn(b)= o E0eτ (n) Mn(c) , ∀a, b, c ∈ (0, 1).
This is also true forFL with the assumption in Theorem 5.2. But, the converse is not
necessarily true. For an illustration, recall the example in Remark 1.7. It has been proved that E0τe (n) Mn(a) 1 λn 1 ξn + n2, ∀a ∈ (0, 1).
By Lemma A.1, one may compute
Var0τe (n) 1 = 1 ξ2 n and Var1eτ (n) Mn(a)≥ Mn(a)−1 X i=1 1 Kn(i, i + 1)πn(i) i X `=1 πn(`)E`eτ (n) i+1 n 4.
Along with the fact Var0eτ
(n)
i ≤ (E0eτ
(n) i )
2, we may conclude from the above computations
that Var0eτ (n) Mn(a) ξ
−2
n + n4for alla ∈ (0, 1). By Theorem 5.1, this implies that the family
FL
c has no total variation cutoff. It has been shown in Remark 1.7 that ifn2ξn → 0, then
E0eτ
(n) Mn(a)∼ ξ
−1
Remark 5.5. Let vn(a) and wn(a) be the constants in Theorems 5.1 and 5.2. It is
remarkable that ifδ = infi,nKn(i, i) > 0, thenδvn2(a) ≤ w2n(a) ≤ vn2(a)for alla ∈ (0, 1). To
see this, we letβ1(n), ..., βM(n)
n be the eigenvalues of the submatrix ofI − Kn indexed by
0, ..., Mn(a) − 1. By Lemma 2.1,β (n)
i > 0for alliand
v2n(a) = Mn(a) X i=1 1 (βi(n))2, w 2 n(a) = Mn(a) X i=1 1 − βi(n) (βi(n))2 . Clearly, w2
n(a) ≤ vn2(a). For the lower bound ofw2n(a), set K (δ)
n = (Kn− δI)/(1 − δ).
Note thatKn(δ) is also a stochastic matrix and the submatrix ofI − K (δ)
n indexed by
0, ..., Mn(a) − 1has eigenvaluesβ (n)
1 /(1 − δ), ..., β (n)
Mn(a)/(1 − δ). By Remark 2.1, we have
(1 − δ) Mn(a) X i=1 1 (β(n)i )2 ≥ Mn(a) X i=1 1 βi(n)
and this impliesw2n(a) ≥ δvn2(a).
Remark 5.6. Note that, in Theorems 5.1 and 5.2, if one choosesE0eτ
(n)
Mn(a)andE0τ
(n) Mn(a)
as the cutoff times, the square roots of Var0eτ (n)
Mn(a)and Var0τ
(n)
Mn(a)are no longer suitable
for the respective cutoff windows. This is very different from the conclusion in Theorem 1.4 and we refer the reader to Example 5.1 for an illustration of this observation. Remark 5.7. By Theorems 5.1 and 5.2, if, based on the assumption ofπn(0) → 0,FcL
(resp. FL) has a total variation cutoff with cutoff timet
n(resp. tn→ ∞), then E0eτ (n) Mn(a)∼ tn (resp.E0τ (n) Mn(a)∼ tn), ∀a ∈ (0, 1). This implies E0eτ (n) Mn∼ tn (resp.E0τ (n) Mn ∼ tn),
for any sequenceMn satisfyinginfnπn([0, Mn]) > 0andinfnπn([Mn, n]) > 0. Letλn(a),
vn(a)andwn(a)be the quantities in Theorems 5.1 and 5.2. As it is easy to check that
λn(a) ≤ λn(b), vn(a) ≤ vn(b), wn(a) ≤ wn(b), ∀0 < a < b < 1,
one may relax the selection of stateMn(a), which generally requires detailed information
ofπn, in Theorems 5.1 and 5.2 to any sequenceMnwhich satisfiesinfnπn([0, Mn]) > 0
andinfnπn([Mn, n]) > 0. The following theorem summarizes the above discussions.
Theorem 5.3. LetF , Fc andλn(a), un(a), vn(a), wn(a)be as in Theorems 5.1 and 5.2.
Suppose thatπn(0) → 0and letan∈ (0, 1)be any sequence satisfying
inf
n≥1an> 0, supn≥1an< 1. (5.2)
(1) ForFc, the following are equivalent.
(1-1) FL
c has a total variation cutoff.
(1-2) un(an)/vn(an) → ∞for any sequencean satisfying (5.2).
(1-3) un(an)λn(an) → ∞for any sequencean satisfying (5.2).
Further, if (1-2) or (1-3) holds, thenFL
c has cutoff time (un(an))∞n=1for any
se-quenceansatisfying (5.2).
(2) ForF, assume thatinfi,nKn(i, i) > 0and there is a sequencean satisfying (5.2)
(2-1) FL has a total variation cutoff.
(2-2) un(an)/wn(an) → ∞for any sequenceansatisfying (5.2).
(2-3) un(an)λn(an) → ∞for any sequencean satisfying (5.2).
Further, if (2-2) or (2-3) holds, thenFL has cutoff time (u
n(an))∞n=1for any
se-quenceansatisfying (5.2).
The next corollary, of which proof is lengthy and addressed in Section 7, provides a way of selecting cutoff windows.
Corollary 5.4. Let Fc, un(a), vn(a)be as in Theorem 5.1. If FcL has a total variation
cutoff andbn> 0is a sequence satisfying
bn= o(un(a)), vn(a) = O(bn), ∀a ∈ (0, 1),
thenFL
c has a(un(a), bn)total variation cutoff. The above statement is also true forFL
under the assumption ofinfn,iKn(i, i) > 0andinfnbn > 0and the replacement ofvn(a)
bywn(a)in Theorem 5.2.
Example 5.1. LetF = (Xn, Kn, πn)∞n=1be a family of birth and death chains for which
Xn = {0, 1, ..., n},πn(i) = 2−n ni and
Kn(i, i + 1) = 1 −ni, Kn(i + 1, i) = i+1n fori 6= Mn,
Kn(Mn, Mn+ 1) = cn 1 − Mnn , Kn(Mn, Mn) = (1 − cn) 1 −Mnn , Kn(Mn+ 1, Mn) = cn(Mn+1) n , Kn(Mn+ 1, Mn+ 1) = (1−cn)(Mn+1) n ,
wherecn∈ (0, 1)andMn ∈ Xn is a state satisfyingπn([0, Mn]) ≥ 1/4andπn([Mn, n]) ≥
3/4. LetFc be the family associated withF andeτ
(n)
i be the first hitting time to stateiof
thenth chain inFc. We will also useMn(a)witha ∈ (0, 1)to denote a state satisfying
πn([0, Mn(a)]) ≥ aandπn([Mn(a), n]) ≥ 1 − a. Whencn = 1,(Xn, Kn, πn)is the Ehrenfest
chain on{0, 1, ..., n}. The spectral information of the Ehrenfest chain is well-studied and it is easy to derive by Lemma 2.2 that
E0eτ (n) bn/2c =
1
4n log n + O(n), Var0τe
(n) bn/2c n
2.
One may use Stirling’s formula to show that, for0 < a < b < 1,
n 2 − Mn(a) √ n, πn(i) 1 √
n uniformly forMn(a) ≤ i ≤ Mn(b).
By Lemmas A.1, 2.2 and 7.1, this implies that, fora ∈ (0, 1),
E0eτ
(n) Mn(a)=
1
4n log n + O(n), Var0τe
(n) Mn(a) n
2. (5.3)
Whencnis small,(Xn, Kn, πn)is the modification of the Ehrenfest chain with bottleneck
between statesMnandMn+ 1. In the following, we will discuss the total variation cutoff
and the cutoff window ofFL
c whencnis small.
First, we consider the total variation cutoff ofFL
c. By Lemma A.1 and (5.3), one can
show without difficulty that, fora ∈ (0, 1/2),
E0eτ (n) Mn(a)=
1
4n log n + O(n), Var0τe
(n) Mn(a) n 2, (5.4) and, fora ∈ (1/2, 1), E0τe (n) Mn(a)= 1 4n log n + O(n) + 1 + o(1) 2cnπn(Mn) , Var0τe (n) Mn(a) n 2+ n c2 n , (5.5)
where πn(Mn) 1/
√
n. By Theorem 5.1,FL
c has a total variation cutoff if and only if
cn
√
n log n → ∞.
Next, we discuss the cutoff window ofFL
c . Assume thatcn
√
n log n → ∞. By Corollary 5.4 and Equations (5.4) and (5.5), FL
c has a ( 1
4n log n, max{
√
n/cn, n})total variation
cutoff. We will prove that the window is optimal whencn
√ n → 0. Supposecn √ n → 0 and set sn= E0τe (n) Mn, tn= E0eτ (n) Mn+1, a 2 n=Var (n) 0 τe (n) Mn, b 2 n=Var (n) 0 eτ (n) Mn+1.
LetTn,cTV(0, )be the total variation mixing time of thenth chain inF
L c and recall (7.2) in the following Tn,(c)TV(0, ) ≤ E0eτ (n) i + q (1−δ δ )Var0(τe (n) i ) for = δ + πn([i + 1, n]) ≥ E0eτ (n) i − q (1−δδ )Var0(τe (n) i ) for = δ − πn([0, i − 1]) .
In the first inequality, the replacement ofi = Mnandδ = 1/8implies
Tn,cTV(0, 7/8) ≤ sn+ 3an.
In the second inequality, the replacement ofi = Mn+ 1andδ = 3/8gives
Tn,cTV(0, 1/8) ≥ tn−
4 5bn.
These two inequalities yield
Tn,cTV(0, 1/8) − Tn,cTV(0, 7/8) ≥ EMnτe
(n)
Mn+1− 3an−
4 5bn.
Under the assumption thatcn
√
n → 0, one may compute using Lemma A.1 that
an n, bn∼ EMneτ (n) Mn+1 √ n cn = n cn √ n. Consequently, whencn √
n → 0, the cutoff window can be Var0eτ (n)
Mn(a)for anya ∈ (1/4, 1)
but not fora ∈ (0, 1/4). Similar observation also happens inFR c .
We would like to point out an interesting observation arising from the bottleneck effect in this example. Compared with the casecn = 1for alln, when cn is of order
bigger than1/√n,FL
c has a cutoff with the same cutoff time and window. Whencn is
of order between1/√nand1/√n log n,FL
c has a cutoff with the same cutoff time but
different (larger) cutoff window. Whencn is of order smaller than1/
√
n log n, the cutoff ofFL
c disappears.
6
Comparison of total variation cutoffs
In this section, we make a comparison of cutoffs introduced in Sections 3 and 5. To avoid confusion, we useF , Fc to denote families of birth and death chains without initial
states specified and letFL, FL
c andFR, FcRbe families of chains started at respectively
left and right boundary states. The following theorem is an immediate corollary of Theorems 5.1 and 5.2 and the proof is given in the end of this section.
Theorem 6.1. LetF = (Xn, Kn, πn)∞n=1be a family of irreducible birth and death chains
withXn= {0, ..., n}andFc be the family of continuous time chains associated withF.
For any sequenceS = (xn)∞n=1withxn∈ Xn, letFS, FcS be the families of chains inF , Fc
(1) IfFL
c andFcRhave a total variation cutoff with cutoff timern andsn, thenFchas a
maximum total variation cutoff with cutoff timetn, wheretn= max{rn, sn}.
(2) LetMn ∈ Xn be a sequence of states satisfying
inf
n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0
and letS = (xn)∞n=1, wherexn∈ {0, n}is a state such that
maxnE0τe (n) Mn, Enτe (n) Mn o = Exneτ (n) Mn
and τei(n) is the first hitting time to state i of the nth chain in Fc. If Fc has a
maximum total variation cutoff with cutoff timetn, thenFcS has a total variation
cutoff with cutoff timetn. In particular,FcS has a(Exneτ
(n)
Mn, bn)total variation cutoff
withb2 n= max{Var0eτ (n) Mn,Varnτe (n) Mn}.
The above statements also apply forFunder the assumptioninfn,iKn(i, i) > 0.
Remark 6.1. LetFc,eτ (n)
i , Mn(a)be as in Theorem 5.1. By Theorem 6.1(2) and Remark
5.4, ifFchas a maximum total variation cutoff, then
EMn(a)eτ (n) Mn(b)= o maxnE0eτ (n) Mn(c), Enτe (n) Mn(c) o , ∀a, b, c ∈ (0, 1).
The following example gives counterexamples to the converse of (1) and (2) in Theorem 6.1.
Example 6.1. Consider the familyF = (Xn, Kn, πn)∞n=1, whereXn= {0, 1, ..., n}and
Kn(i, i + 1) = 1 −2ni , ∀0 ≤ i < n, i 6= in, Kn(i + 1, i) = i+12n, ∀0 ≤ i < n − 1, i 6= in, Kn(n, n − 1) = 1, Kn(in, in+ 1) = cn(1 −2nin), Kn(in+ 1, in) = cnin2n+1, Kn(in, in) = (1 − cn)(1 −2nin), Kn(in+ 1, in+ 1) = (1 − cn)in2n+1,
with0 ≤ in< nandcn ∈ [0, 1], and
πn(i) = 21−2n 2n i , ∀0 ≤ i < n, πn(n) = 2−2n 2n n .
As before, we use Mn(a) to denote a state in Xn satisfying πn([0, Mn(a)]) ≥ a and
πn([Mn(a), n]) ≥ 1 − aand letτe
(n)
i be the first hitting time to stateiof the continuous
time chain associated with(Xn, Kn, πn). Let0 < λn,1< λn,2< · · · < λn,nbe eigenvalues
ofI − Kn. It follows immediately from the central limit theorem that
n − Mn(a)
√
n, ∀a ∈ (0, 1). (6.1)
In what follows, we discuss the total variation cutoffs ofFc,FcL andF R
c with specific
cn andin.
First, assume thatcn = 1for alln. In this setting, the chain(Xn, Kn, πn)is exactly
the collapsed chain of the Ehrenfest model on{0, 1, ..., 2n}obtained by combining states
{i, 2n − i}into a new state for0 ≤ i < n. The spectral information of the Ehrenfest model is well-studied and this implies
λn,i=
2i
By Theorem 1.1,Fc has a maximum separation cutoff with cutoff time12n log nand, thus,
has a maximum total variation cutoff. A simple computation with the Stirling formula gives
πn(i)
1 √
n, uniformly forMn(a) ≤ i ≤ n.
By Lemma A.1, this implies that, fora ∈ (0, 1),
Enτe
(n)
Mn(a) n, VarnτeMn(a) n
2
,
and, by Theorem 1.3, we haveE0eτ
(n) Mn(a)∼
1
2n log nfor anya ∈ (0, 1). As a consequence
of Theorems 5.1 and 6.1(2),FR
c has no total variation cutoff, butFcLhas with cutoff time 1
2n log n. Furthermore, by Theorem 1.4(1), the total variation cutoff time forFccan be 1
2n log n. This gives a counterexample to the converse of Theorem 6.1(1).
Next, we consider the casen − in= o(
√
n)andcnis small. The assumption of smallcn
denotes a bottleneck between statesinandin+ 1. Under the assumptionn − in= o(
√ n), (6.1) implies that, fora ∈ (0, 1), bothE0eτ
(n)
Mn(a)and Var0τe
(n)
Mn(a)remain the same as in the
casecn = 1. This implies thatFcLhas a total variation cutoff with cutoff time 1 2n log n.
For the cutoff ofFR
c , one may compute using the formula in Lemma A.1 that, for any
a ∈ (0, 1), Eneτ (n) Mn(a) n + n − in cn , VarneτMn(a) n +n − in cn 2 .
Consequently, Theorem 5.1 implies thatFR
c has no cutoff in total variation. Moreover,
Theorem 1.4 implies that if(n−in)/cn= o(n log n), thenFchas a maximum total variation
cutoff. Ifn log n = O((n − in)/cn), thenFchas no maximum total variation cutoff, which
gives a counterexample to the converse of Theorem 6.1(2).
The next theorem provides more information on the comparison of cutoffs and should be regarded as a complement to Theorem 6.1.
Theorem 6.2. Let F = {(Xn, Kn, πn)∞n=1 be a family of birth and death chains with
Xn = {0, 1, ..., n} and Fc be the family of continuous time chains associated with F.
Suppose thatπn({0, n}) → 0and, in total variation,FcLhas a cutoff with cutoff timetn
but no subsequence ofFR
c has a cutoff. LetMn be a state inXnand set
R = lim sup n→∞ Eneτ (n) Mn tn . (6.2)
Then, the following are equivalent.
(1) Fchas a maximum total variation cutoff. In particular,tn is a cutoff time.
(2) R = 0for some sequence(Mn)∞n=1satisfying
inf
n≥1πn([0, Mn]) > 0, n≥1inf πn([Mn, n]) > 0. (6.3)
(3) R = 0for any sequence(Mn)∞n=1satisfying (6.3).
The above statement also holds forFprovidedinfn,iKn(i, i) > 0.
Remark 6.2. Consider the familyF in Theorem 6.2. Suppose thatπn(0) → 0andFcL
has a total variation cutoff with cutoff timetn. LetRbe the constant in (6.2), where
Mn is a sequence satisfying (6.3). Fora ∈ (0, 1), let0 ≤ Mn(a) ≤ nbe a state satisfying
to see thatEMn(a)τe
(n)
Mn(b)= o(tn)for all0 < a < b < 1. Further, one may use the following
inequality, Ejeτ (n) i ≤ πn([j + 1, n]) πn([0, i]) E ieτ (n) j , ∀0 ≤ i < j ≤ n,
which can be derived using Lemma A.1, to getEMn(b)τe
(n)
Mn(a)= o(tn)for all0 < a < b < 1.
This implies, for0 < a ≤ infnπn([0, Mn])andsupnπn([Mn, n]) ≤ b < 1,
lim sup n→∞ Eneτ (n) Mn(b) tn ≤ R ≤ lim sup n→∞ Eneτ (n) Mn(a) tn ≤ lim sup n→∞ Eneτ (n) Mn(b) tn + lim sup n→∞ EMn(b)eτ (n) Mn(a) tn = lim sup n→∞ Eneτ (n) Mn(b) tn . As a consequence, we obtain R = lim sup n→∞ Eneτ (n) Mn(a) tn ∀0 < a < 1. (6.4)
In particular, the limitRis independent of the choice of(Mn)∞n=1subject to (6.3).
Note that the conclusion in (6.4) also applies for the discrete time case with the further assumptioninfi,nKn(i, i) > 0. In detail, the proof for the casetn→ ∞is similar
to the continuous time case. Iftn has a bounded subsequence, saytkn, then, by Remark
5.3,E0τ (kn)
Mkn(a)= O(1)for anya ∈ (0, 1). As a consequence of the observationE0τ
(n) i ≥ i,
one hasMkn(a) = O(1)and, then,Enτ
(kn)
Mkn(a)≥ n − Mkn(a) → ∞for alla ∈ (0, 1). This
leads to lim sup n→∞ Enτ (n) Mn(a) tn = ∞, ∀a ∈ (0, 1), and R ≥ lim sup n→∞ Enτ (kn) Mkn(a) tkn = ∞, ∀ sup n πn([0, Mn]) < a < 1, as desired.
It is worthwhile to remark that, in the above discussions,lim supcan be replaced by
limprovided thatEnτe (n)
Mn/tn andEnτ
(n)
Mn/tn converge.
Proof of Theorem 6.2. By Remark 6.2, it is obvious that (2) and (3) are equivalent and the choice ofMn can be restricted toMn(a), a state such thatπn([0, Mn(a)]) ≥ aand
πn([Mn(a), n]) ≥ 1 − a.
We first consider the continuous time case. SinceFL
c has a total variation cutoff with
cutoff timetn, Theorem 5.1 implies
E0eτ (n) Mn(a)∼ tn, Var0eτ (n) Mn(a)= o(t 2 n), ∀a ∈ (0, 1). (6.5)
For (2)⇒(1), assume that R = 0 with Mn = Mn(a)for somea ∈ (0, 1). This implies
Eneτ (n)
Mn(a) = o(tn) and, then, Varneτ
(n)
Mn(a) = o(t
2
n) using the fact Varneτ (n)
i ≤ (Eneτ (n) i )2.
Combining this observation with (6.5) yields
r maxnVar0τe (n) Mn(a),Varneτ (n) Mn(a) o = omaxnE0τe (n) Mn(a), Eneτ (n) Mn(a) o . (6.6) By Theorem 1.4,Fc has a maximum total variation cutoff with cutoff timetn.
For (1)⇒(3), we prove the equivalent implication by assuming thatR > 0for some sequence(Mn)∞n=1satisfying (6.3). Note that one may choose a subsequence(kn)∞n=1
such that lim n→∞ Ekneτ (kn) Mkn tkn = R > 0. (6.7)
For the subfamily ofFL
c indexed by(kn)∞n=1, Remark 6.2 implies that the limit in (6.7)
also holds forMkn= Mkn(a)witha ∈ (0, 1). Further, as the subfamily ofF
R
c indexed by
(kn)∞n=1is assumed to have no total variation cutoff, we may refine, by Theorem 5.1, the
selection ofkn such that
q Varkneτ (kn) Mkn(a) Eknτe (kn) Mkn(a), tkn= O Ekneτ (kn) Mkn(a) , (6.8)
for somea ∈ (0, 1). Combining (6.5) with the above discussion leads to
r maxnVar0τe (kn) Mkn(a),Varknτe (kn) Mkn(a) o maxnE0τe (kn) Mkn(a), Ekneτ (kn) Mkn(a) o ,
for somea ∈ (0, 1). By Theorem 1.4, the subfamily ofFcindexed by(kn)has no maximum
total variation cutoff.
Next, we consider the discrete time case. For (2)⇒(1), assume that R = 0 with
Mn = Mn(a0)for somea0∈ (0, 1). This impliesEnτ (n)
Mn(a0)= o(tn)and Varnτ
(n) Mn(a0)= o(t 2 n). Observe that E0τ (n) Mn(a)+ Enτ (n) Mn(a)≥ n, ∀a ∈ (0, 1). (6.9)
By Remark 5.3, (6.9) implies tn → ∞. Otherwise, if ln is a subsequence such that
tln is bounded, then E0τ
(ln)
Mln(a0)(≥ Mln(a
0)) is bounded, which implies
Elnτ (ln) Mln(a0) ≥ ln− Mln(a 0) → ∞and then ∞ = lim inf n→∞ ln tln ≤ lim sup n→∞ 2Elnτ (ln) Mln(a0) tln ≤ 2R = 0,
a contradiction. Using Theorem 5.2, one may derive a discrete time version of (6.5) and (6.6). As a consequence of Theorem 1.4,F has a maximum total variation cutoff with cutoff timetn.
For (1)⇒(3), we assume the inverse of (3) thatR > 0for some sequenceMnsatisfying
(6.3). By Remark 6.2, one may select a subsequence`n such that
lim n→∞ E`nτ (`n) M`n(a) t`n = R > 0, ∀a ∈ (0, 1).
Consider the following two refinements of`nsuch that
Case 1:t`n → ∞.
Case 2:t`n = O(1).
The proof of Case 1 is the same as the continuous time case. In Case 2, since the subfamily ofFLindexed by(`
n)has a cutoff with cutoff timet`n, Remark 5.3 implies that
E0τ (`n)
M`n(a)= O(1), Var0τ (`n)
M`n(a)= O(1), ∀a ∈ (0, 1).
By (6.9), we haveE`nτ
(`n)
M`n(a)→ ∞for anya ∈ (0, 1)and, by Theorem 5.2, we may further
refine`n such that the discrete time version of (6.8) holds for somea ∈ (0, 1)with the
replacement ofkn by`n. Consequently, Theorem 1.4 implies that the subfamily of F
The next theorem is a special version of Theorem 6.1 which identifies two different cutoffs discussed in this section.
Theorem 6.3. LetF = (Xn, Kn, πn)∞n=1be a family of irreducible birth and death chains
withXn= {0, ..., n}andFcbe the families of continuous time chains associated withF.
Assume thatKn(i, j) = Kn(n − i, n − j)for alli, j ∈ Xn andn ≥ 1.
(1) FL
c has a total variation cutoff with cutoff timetn if and only ifFchas a maximum
total variation cutoff with cutoff timetn.
(2) Under the assumption thatinfn,iKn(i, i) > 0,FLhas a total variation cutoff with
cutoff timetnif and only ifFhas a maximum total variation cutoff with cutoff time
tn.
Proof of Theorem 6.1(Continuous time case). As before, we useeτi(n)to denote the first hitting time to stateiof thenth chain inFcand use the notationMn(a)witha ∈ (0, 1)to
denote a state inXnsatisfyingπn([0, Mn(a)]) ≥ aandπn([Mn(a), n]) ≥ 1 − a.
For (1), assume thatFL
c, FcRhave total variation cutoffs with cutoff timesrn, sn. By
Theorem 5.1, we have q Var0τe (n) Mn(1/2)= o E0eτ (n) Mn(1/2) , E0eτ (n) Mn(1/2)∼ rn, and q Varneτ (n) Mn(1/2)= o Eneτ (n) Mn(1/2) , Eneτ (n) Mn(1/2)∼ sn.
Clearly, this implies
r maxnVar0eτ (n) Mn(1/2),Varneτ (n) Mn(1/2) o = omaxnE0eτ (n) Mn(1/2), Eneτ (n) Mn(1/2) o and maxnE0eτ (n) Mn(1/2), Eneτ (n) Mn(1/2) o ∼ max{rn, sn} = tn.
By Theorem 1.4,Fc has a maximum total variation cutoff with cutoff timetn.
For (2), letF = (Xb n, bKn,bπn)∞n=1be a family given by
b
Kn= Kn, bπn= πn ifxn= 0,
and
b
Kn(i, j) = Kn(n − i, n − j), πbn(i) = πn(n − i), ∀i, j ∈ Xn ifxn = n.
LetFbc be the family of continuous time chains associated withFb. Suppose thatFchas
a maximum total variation cutoff with cutoff timetn. It is obvious thatFbc also has a
maximum total variation cutoff with cutoff time tn and, to show thatFcS has a total
variation cutoff with cutoff timetn, it is equivalent to prove thatFbcL has a total variation
cutoff with cutoff timetn.
Letbτi(n)be the first hitting time to stateiof the continuous time chain associated with(Xn, bKn,bπn)and setMcn be a state defined by
c Mn = ( Mn ifxn= 0 n − Mn ifxn= n .
We useMcn(a)to denote a state such that
b
By Theorem 1.4, the total variation cutoff ofFbcwith cutoff timetn implies tn ∼ max n E0τb (n) c Mn, Enb τ(n) c Mn o = E0bτ (n) c Mn
and, for anya ∈ (0, 1),
r maxnVar0τb (n) c Mn(a) ,Varnbτ (n) c Mn(a) o = o(tn) = o E0bτ (n) c Mn . (6.10) As a result of Lemma 7.1 and (6.10), we have, for0 < b < a < 1,
EMcn(b)τb (n) c Mn(a) = O r Var c Mn(b)τb (n) c Mn(a) = oE0τb (n) c Mn , which leads to E0bτ (n) c Mn(a)∼ E0b τ(n) c Mn , ∀a ∈ (0, 1).
Applying the last identity to (6.10) yields
r Var0bτ (n) c Mn(a) = oE0τb (n) c Mn(a) , ∀a ∈ (0, 1).
By Theorem 5.1,FbcLhas a total variation cutoff with cutoff timetn. The precise
descrip-tion of the cutoff time and window is given by Theorem 1.4, Corollary 5.4 and Remark 1.5.
Proof of Theorem 6.1(Discrete time case). We useτi(n)to denote the first hitting time to stateiof thenth chain inF andMn(a)for a state inXnsatisfyingπn([0, Mn(a)]) ≥ aand
πn([Mn(a), n]) ≥ 1 − a.
For (1), assume thatFL, FRhave cutoffs with respective cutoff timesr
n, sn. Given
an increasing sequenceK = (kn)∞n=1in{1, 2, ...}, letF (K)be the family of chains inF
indexed by the sequenceK. By Proposition 2.1 in [7], to proveF has a maximum total variation cutoff, it suffices to show that, for any increasing sequence of positive integers, there is a subsequence, sayK, such that F (K)has a maximum total variation cutoff. Note that, by Remark 5.3,rn+ snmust tend to infinity. This implies thatKcan be chosen
to satisfy one of the following cases.
Case 1:rkn→ ∞andskn→ ∞.
Case 2:rkn→ ∞andskn= O(1).
Case 3:rkn= O(1)andskn → ∞.
The proof for Case 1 is the same as the continuous time case. The proofs of Case 2 and Case 3 are similar and we discuss Case 2, here. By Theorem 5.2 and Remark 5.3, the cutoffs ofFL, FRimply that, fora ∈ (0, 1),
E0τ (kn) Mkn(a)∼ rkn, q Var0τ (kn) Mkn(a)= o(rkn), and q Varknτ (kn) Mkn(a)≤ Eknτ (kn) Mkn(a)= O(1).
This implies, fora ∈ (0, 1),
r maxnVar0τ (kn) Mkn(a),Varknτ (kn) Mkn(a) o = omaxnE0τ (kn) Mkn(a), Eknτ (kn) Mkn(a) o and maxnE0τ (kn) Mkn(a), Eknτ (kn) Mkn(a) o ∼ max{rkn, skn} = tkn.
