6 Comparison of total variation cutoffs





≤ E0eτ_i⁽ⁿ⁾+ q

(^1−δ_δ )Var₀(τe_i⁽ⁿ⁾) for = δ + π_n([i + 1, n])

≥ E0eτ_i⁽ⁿ⁾−q

(_1−δ^δ )Var₀(τe_i⁽ⁿ⁾) for = δ − π_n([0, i − 1]) .

In the first inequality, the replacement ofi = M_nandδ = 1/8implies T_n,^c_TV(0, 7/8) ≤ s_n+ 3a_n.

In the second inequality, the replacement ofi = M_n+ 1andδ = 3/8gives T_n,^c_TV(0, 1/8) ≥ tn−4

5bn. These two inequalities yield

T_n,^c_TV(0, 1/8) − T_n,^c_TV(0, 7/8) ≥ EMnτe_M⁽ⁿ⁾

n+1− 3a_n−4 5b_n. Under the assumption thatc_n√

n → 0, one may compute using Lemma A.1 that

an  n, bn∼ EM_neτ_M⁽ⁿ⁾

n+1

√n cn

= n

cn

√n.

Consequently, whencn

√n → 0, the cutoff window can be Var0eτ_M⁽ⁿ⁾

n(a)for anya ∈ (1/4, 1) but not fora ∈ (0, 1/4). Similar observation also happens inF_c^R.

We would like to point out an interesting observation arising from the bottleneck effect in this example. Compared with the casec_n = 1for alln, when c_n is of order bigger than1/√

n,F_c^L has a cutoff with the same cutoff time and window. Whencn is of order between1/√

nand1/√

n log n,F_c^L has a cutoff with the same cutoff time but different (larger) cutoff window. Whencn is of order smaller than1/√

n log n, the cutoff ofF_c^L 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 letF^L, F_c^LandF^R, F_c^Rbe 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 = (X_n, K_n, π_n)^∞_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, letF^S, F_c^S be the families of chains inF , Fc

for which thenth chain started atxn. Assume thatπn({0, n}) → 0.

(1) IfF_c^LandF_c^Rhave a total variation cutoff with cutoff timer_n ands_n, thenFchas a maximum total variation cutoff with cutoff timet_n, wheret_n= max{r_n, s_n}. (2) LetMn ∈ Xn be a sequence of states satisfying

inf

n≥1πn([0, Mn]) > 0, inf

n≥1πn([Mn, n]) > 0 and letS = (x_n)^∞_n=1, wherex_n∈ {0, n}is a state such that

maxn E0τe_M⁽ⁿ⁾

n, Enτe_M⁽ⁿ⁾

n

o

= Exneτ_M⁽ⁿ⁾

n

and τe_i⁽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 timet_n, thenF_c^S has a total variation cutoff with cutoff timetn. In particular,F_c^S has a(E^xneτ_M⁽ⁿ⁾

n, bn)total variation cutoff withb²_n= max{Var0eτ_M⁽ⁿ⁾

n,Varnτe_M⁽ⁿ⁾

n}.

The above statements also apply forFunder the assumptioninfn,iKn(i, i) > 0.

Remark 6.1. LetFc,eτ_i⁽ⁿ⁾, 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_M⁽ⁿ⁾

n(b)= o maxn

E⁰eτ_M⁽ⁿ⁾

n(c), Eⁿτe_M⁽ⁿ⁾

n(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 = (X_n, K_n, π_n)^∞_n=1, whereX_n= {0, 1, ..., n}and















Kn(i, i + 1) = 1 −_2nⁱ , ∀0 ≤ i < n, i 6= in,

K_n(i + 1, i) = ⁱ⁺¹_2n, ∀0 ≤ i < n − 1, i 6= in, K_n(n, n − 1) = 1, Kn(in, in+ 1) = cn(1 −_2nⁱⁿ), Kn(in+ 1, in) = cni_n+1

2n ,

Kn(in, in) = (1 − cn)(1 −_2nⁱⁿ), Kn(in+ 1, in+ 1) = (1 − cn)ⁱⁿ_2n⁺¹, with0 ≤ in< nandcn ∈ [0, 1], and

πn(i) = 2¹⁻²ⁿ2n i



, ∀0 ≤ i < n, πn(n) = 2⁻²ⁿ2n n

 .

As before, we use M_n(a) to denote a state in Xn satisfying π_n([0, M_n(a)]) ≥ a and π_n([M_n(a), n]) ≥ 1 − aand letτe_i⁽ⁿ⁾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 − M_n(a) √

n, ∀a ∈ (0, 1). (6.1)

In what follows, we discuss the total variation cutoffs ofFc,F_c^L andF_c^Rwith 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

n, ∀1 ≤ i ≤ n.

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_M⁽ⁿ⁾

n(a) n, VarnτeM_n(a) n², and, by Theorem 1.3, we haveE⁰eτ_M⁽ⁿ⁾

n(a)∼¹₂n log nfor anya ∈ (0, 1). As a consequence of Theorems 5.1 and 6.1(2),F_c^Rhas no total variation cutoff, butF_c^Lhas with cutoff time

1

2n log n. Furthermore, by Theorem 1.4(1), the total variation cutoff time forF_ccan be

1

2n log n. This gives a counterexample to the converse of Theorem 6.1(1).

Next, we consider the casen − i_n= o(√

n)andc_nis small. The assumption of smallc_n denotes a bottleneck between statesinandin+ 1. Under the assumptionn − in= o(√

n), (6.1) implies that, fora ∈ (0, 1), bothE⁰eτ_M⁽ⁿ⁾

n(a)and Var0τe_M⁽ⁿ⁾

n(a)remain the same as in the casecn = 1. This implies thatF_c^Lhas a total variation cutoff with cutoff time ¹₂n log n. For the cutoff ofF_c^R, one may compute using the formula in Lemma A.1 that, for any a ∈ (0, 1),

Eⁿeτ_M⁽ⁿ⁾

n(a) n +n − in

c_n , Varneτ_Mn(a)



n +n − in

c_n

² .

Consequently, Theorem 5.1 implies thatF_c^Rhas 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,F_c^Lhas a cutoff with cutoff timetn

but no subsequence ofF_c^Rhas a cutoff. LetM_n be a state inXnand set

R = lim sup

n→∞

Eⁿeτ_M⁽ⁿ⁾

n

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(M_n)^∞_n=1satisfying

n≥1inf πn([0, Mn]) > 0, inf

n≥1πn([Mn, n]) > 0. (6.3) (3) R = 0for any sequence(M_n)^∞_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 thatπ_n(0) → 0andF_c^L 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 πn([0, Mn(a)]) ≥ aandπn([Mn(a), n]) ≥ 1 − a. By Theorem 5.1 and Remark 5.4, it is easy

to see thatEMn(a)τe_M⁽ⁿ⁾ As a consequence, we obtain

R = lim sup

n→∞

Eⁿeτ_M⁽ⁿ⁾

n(a)

t_n ∀0 < a < 1. (6.4)

In particular, the limitRis independent of the choice of(M_n)^∞_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, saytk_n, then, by Remark 5.3,E⁰τ_M^(kn)

It is worthwhile to remark that, in the above discussions,lim supcan be replaced by limprovided thatEⁿτe_M⁽ⁿ⁾

n/tn andEⁿτ_M⁽ⁿ⁾

n/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. SinceF_c^Lhas a total variation cutoff with cutoff timetn, Theorem 5.1 implies

E0eτ_M⁽ⁿ⁾ Combining this observation with (6.5) yields

r

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(M_n)^∞_n=1satisfying (6.3). Note that one may choose a subsequence(k_n)^∞_n=1 such that

n→∞lim

E^kneτ_M^(kn)

kn

tk_n

= R > 0. (6.7)

For the subfamily ofF_c^Lindexed by(kn)^∞_n=1, Remark 6.2 implies that the limit in (6.7) also holds forMk_n= Mk_n(a)witha ∈ (0, 1). Further, as the subfamily ofF_c^Rindexed 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_M^(kn)

kn(a) Eknτe_M^(kn)

kn(a), t_kn= O

Ekneτ_M^(kn)

kn(a)

, (6.8)

for somea ∈ (0, 1). Combining (6.5) with the above discussion leads to r

maxn

Var0τe_M^(kn)

kn(a),Vark_nτe_M^(kn)

kn(a)

o maxn E⁰τe_M^(kn)

kn(a), E^kneτ_M^(kn)

kn(a)

o ,

for somea ∈ (0, 1). By Theorem 1.4, the subfamily ofFcindexed by(k_n)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_M⁽ⁿ⁾

n(a⁰)= o(tn)and Varnτ_M⁽ⁿ⁾

n(a⁰)= o(t²_n). Observe that

E⁰τ_M⁽ⁿ⁾

n(a)+ Eⁿτ_M⁽ⁿ⁾

n(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_M^(ln)

ln(a⁰)(≥ Mln(a⁰)) is bounded, which implies E^lnτ_M^(ln)

ln(a⁰) ≥ l_n− Mln(a⁰) → ∞and then

∞ = lim inf

n→∞

l_n tl_n

≤ lim sup

n→∞

2Elnτ_M^(ln)

ln(a⁰)

tl_n

≤ 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 sequenceM_nsatisfying (6.3). By Remark 6.2, one may select a subsequence`_n such that

n→∞lim

E^{`</sup>nτ<sub>M</sub><sup>(`n)}

`n(a)

t`_n

= R > 0, ∀a ∈ (0, 1).

Consider the following two refinements of`<sub>n</sub>such that Case 1:t<sub>`n</sub> → ∞.

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 ofF^Lindexed by(`n)has a cutoff with cutoff timet`_n, Remark 5.3 implies that

E⁰τ_M^(`n)

`n(a)= O(1), Var0τ<sub>M</sub><sup>(`n)</sup>

`n(a)= O(1), ∀a ∈ (0, 1).

By (6.9), we haveE^{`</sup>nτ<sub>M</sub><sup>(`n)}

`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 indexed by(`n)(and, hence,F) has no maximum total variation cutoff.

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 thatK_n(i, j) = K_n(n − i, n − j)for alli, j ∈ X_n andn ≥ 1.

(1) F_c^L 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,F^Lhas 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 useeτ_i⁽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 thatF_c^L, F_c^Rhave total variation cutoffs with cutoff timesrn, sn. By Theorem 5.1, we have

q

Var0τe_M⁽ⁿ⁾

n(1/2)= o E⁰eτ_M⁽ⁿ⁾

n(1/2)



, E⁰eτ_M⁽ⁿ⁾

n(1/2)∼ rn, and

q

Varneτ_M⁽ⁿ⁾

n(1/2)= o Eⁿeτ_M⁽ⁿ⁾

n(1/2)



, Eⁿeτ_M⁽ⁿ⁾

n(1/2)∼ sn. Clearly, this implies

r maxn

Var₀eτ_M⁽ⁿ⁾

n(1/2),Var_neτ_M⁽ⁿ⁾

n(1/2)

o= o maxn

E0eτ_M⁽ⁿ⁾

n(1/2), Eneτ_M⁽ⁿ⁾

n(1/2)

o

and

maxn E⁰eτ_M⁽ⁿ⁾

n(1/2), Eⁿeτ_M⁽ⁿ⁾

n(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

Kb_n= K_n, bπ_n= π_n ifx_n= 0, and

Kbn(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 timet_n. It is obvious thatFbc also has a maximum total variation cutoff with cutoff time t_n and, to show thatF_c^S has a total variation cutoff with cutoff timetn, it is equivalent to prove thatFb_c^L has a total variation cutoff with cutoff timetn.

Letbτ_i⁽ⁿ⁾be the first hitting time to stateiof the continuous time chain associated with(Xn, bKn,bπn)and setMcn be a state defined by

Mcn =

(Mn ifxn= 0 n − Mn ifxn= n. We useMc_n(a)to denote a state such that

bπn([0, cMn(a)]) ≥ a, bπn([ cMn(a), n]) ≥ 1 − a.

By Theorem 1.4, the total variation cutoff ofFbcwith cutoff timet_n implies

Applying the last identity to (6.10) yields r

By Theorem 5.1,Fb_c^Lhas a total variation cutoff with cutoff timet_n. 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 useτ_i⁽ⁿ⁾to denote the first hitting time to stateiof thenth chain inF andM_n(a)for a state inXnsatisfyingπ_n([0, M_n(a)]) ≥ aand π_n([M_n(a), n]) ≥ 1 − a.

For (1), assume thatF^L, F^Rhave cutoffs with respective cutoff timesrn, 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,r_n+ s_nmust tend to infinity. This implies thatKcan be chosen to satisfy one of the following cases.

Case 1:rk_n→ ∞andsk_n→ ∞. Case 2:rk_n→ ∞andsk_n= O(1). Case 3:rk_n= O(1)andsk_n → ∞.

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 ofF^L, F^Rimply that, fora ∈ (0, 1),

By Theorem 1.4,F (K)has a maximum total variation cutoff with cutoff timet_kn. For (2), based on the following observation

n ≤ E0τ_i⁽ⁿ⁾+ Enτ_i⁽ⁿ⁾, ∀0 ≤ i ≤ n, we haveE^xnτ_M⁽ⁿ⁾

n → ∞. The remaining proof is similar to the continuous time case and is skipped.

在文檔中 Computing cutoff times of birth and death chains (頁 21-28)