Cutoffs for birth and death chains. Consider a family of irreducible birth and death chains

, where the second inequality requires|Ω| ≥ 2/δ.

2.2. Cutoffs for birth and death chains. Consider a family of irreducible birth and death chains

F = {(Ωn, Kn, πn)|n = 1, 2, ...},

where Ω_n={0, 1, ..., n} and Knhas birth rate p_n,i, death rate q_n,iand holding rate r_n,i. We writeFc,Fδ as families of the corresponding continuous time chains and δ-lazy discrete time chains in F. A criterion on total variation cutoffs for families of birth and death chains was introduced in [15], which say that, for δ ∈ (0, 1), Fc,Fδ have total variation cutoffs if and only if the product of the mixing time and the spectral gap tends to infinity. As the total variation distance is comparable with the separation distance, the authors of [15] identify cutoffs in total variation and separation, where a criterion on separation cutoffs was proposed in [13]. In the recent work [6], the cutoffs forFcandFδ are proved to be equivalent and this leads to the following theorems.

Theorem 2.2. [6, Section 4] Let F = {(Ωn, K_n, π_n)|n = 1, 2, ...} be a family of irreducible birth and death chain with Ω_n={0, 1, ..., n}. For n ≥ 1, let λn,1, ..., λ_n,n be nonzero eigenvalues of I− Kn and set

λn = min

1≤i≤nλn,i, sn= 1 λn,1

+· · · + 1 λn,n

. Then, the following are equivalent.

(1) Fc has a total variation cutoff.

(2) Fδ has a total variation cutoff.

(3) Fc has a total variation precutoff.

(4) Fδ has a total variation precutoff.

(5) T_n,^(c)_TV(ϵ)λn→ ∞ for some ϵ ∈ (0, 1).

(6) T_n,^(δ)_TV(ϵ)λ_n→ ∞ for some ϵ ∈ (0, 1).

(7) snλn→ ∞.

Theorem 2.3. [6, Section 4] Referring to Theorem 2.2, it holds true that, for ϵ, η∈ (0, 1/2) and δ ∈ (0, 1),

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(η).

Further, if there is ϵ₀ ∈ (0, 1/2) such that Tn,^(c)TV(ϵ₀)λ_n or T_n,^(δ)_TV(ϵ₀)λ_n is bounded, then, for any ϵ∈ (0, 1/2) and δ ∈ (0, 1),

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ λ⁻¹n .

2.3. A remark on the precutoff. Note that if there is no cutoff in total variation, the approximation in Theorem 2.3 may fail for ϵ∈ (1/2, 1). This means that, for 0 < ϵ < 1/2 < η < 1, the orders of T_n,^(c)_TV(ϵ) and T_n,^(c)_TV(η) can be different. Consider the following example. For n≥ 3, let Ωn={0, 1, ..., n}, Mn=⌊n/2⌋ and

(2.1)











Kn(i, i + 1) = Kn(i + 1, i) = 1/2 for 0≤ i < n, i ̸= Mn

Kn(Mn, Mn+ 1) = Kn(Mn+ 1, Mn) = ϵn

Kn(0, 0) = Kn(n, n) = 1/2

K_n(M_n, M_n) = K_n(M_n+ 1, M_n+ 1) = 1/2− ϵn

,

with ϵ_n≤ 1/2. Assume that ϵn= o(n⁻²). By Theorem 1.4, we have T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ n/ϵn, ∀ϵ ∈ (0, 1/2), δ ∈ (0, 1).

Next, we consider the δ-lazy discrete time case with δ = 1/2. Let K_n,1/2 = (I + Kn)/2 and K_n^′ be the 1/2-lazy simple random walk on{0, 1, ..., Mn}, that is,







K_n^′(i, i + 1) = K_n^′(i + 1, i) = 1/4, ∀0 ≤ i < Mn

K_n^′(i, i) = 1/2, ∀0 < i < Mn

K_n^′(0, 0) = K_n^′(M_n, M_n) = 3/4

.

For n≥ 3, set

cn= min

0≤i,j≤Mn

K_n,1/2^mn (i, j)

(K_n^′)^mn(i, j), Cn = max

0≤i,j≤Mn

K_n,1/2^mn (i, j) (K_n^′)^mn(i, j). Proposition 2.4. If mn≍ n², then

c_n→ 1, Cn → 1, as n → ∞.

Proof. For ℓ≥ 1, let (i0, i1, ..., iℓ) be a path in{0, 1, ..., Mn}. Note that

∏ℓ k=1

K_n,1/2(i_k−1, i_k)≥

(3/4− ϵn/2 3/4

)ℓ∏ℓ

k=1

K_n^′(i_k−1, i_k).

This implies c_n≥ (1 − 2ϵn/3)^mn∼ 1 as n → ∞. To see an upper bound of Cn, one may use Lemma 4.4 in [15] to conclude that, for 0≤ i ≤ n and ℓ ≥ 0,

{

K_n,1/2^ℓ (i, j)≥ K_n,1/2^ℓ (i, j− 1) ∀1 ≤ j ≤ 0 K_n,1/2^ℓ (i, j)≥ K_n,1/2^ℓ (i, j + 1) ∀i ≤ j < n, and, for 0≤ i ≤ Mn and ℓ≥ 0,

{

(K_n^′)^ℓ(i, j)≥ (Kn^′)^ℓ(i, j− 1) ∀1 ≤ j ≤ i (K_n^′)^ℓ(i, j)≥ (Kn^′)^ℓ(i, j + 1) ∀i ≤ j < Mn

.

By the induction, the above observation implies that, for any probabilities µ, ν on {0, ..., n}, {0, ..., Mn} satisfying µ(i) = ν(i) for 0 ≤ i ≤ Mn,

µK_n,1/2^ℓ (j)≤ ν(Kn^′)^ℓ(j), ∀0 ≤ j ≤ Mn, ℓ≥ 0.

This yields C_n≤ 1 for all n ≥ 3. 

For ϵ∈ (0, 1), let Tn,^′TV(ϵ) be the total variation mixing time for K_n^′. It is well-known that, for ϵ ∈ (0, 1), Tn,^′ TV(ϵ)≍ n². Let d^(1/2)_n,TV, d^′_n,TV be the total variation distance for K_n,1/2, K_n^′. As a consequence of the above discussion, we obtain, for ϵ∈ (0, 1),

lim sup

n→∞ d^(1/2)_n,TV(T_n,^′ _TV(ϵ))≤ 1 2

(

1 + lim sup

n→∞ d^′_n,TV(T_n,^′ _TV(ϵ) )

≤1 + ϵ 2 . Thus, for ϵ∈ (1/2, 1), Tn,^(1/2)TV(ϵ) = O(n²). Note that, for mn= o(n²),

nlim→∞

∑

i≤an

K_n,1/2^mn (0, i) = 1, ∀a > 0.

This yields n²= O(T_n,^(1/2)_TV(ϵ)) for ϵ > 0. The above discussion is also valid for the continuous time case and any δ-lazy discrete time case. We summarizes the results in the following theorem.

Theorem 2.5. LetF = {(Ωn, Kn, πn)|n = 1, 2, ...} be the family of birth and death chains in (2.1) and δ ∈ (0, 1). Suppose that ϵn = o(n⁻²). Then, there is no total variation cutoff forFc andFδ. Furthermore, for ϵ∈ (0, 1/2),

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ n/ϵn, and, for ϵ∈ (1/2, 1),

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ n².

Remark 2.2. Figure 1 displays the total variaton distances of the birth and death chains on{1, 2, ..., 100} with transition matrices K1and K₂ given by





















K1(i, i) = 1/2, for i /∈ {1, 50, 51, 100}

K1(i, i + 1) = K1(i + 1, i) = 1/4, for i < 50 or i > 51 K1(i, i) = 3/4 for k∈ {1, 100}

K1(i, i + 1) = K1(i + 1, i) = 10⁻³ for k = 50 K1(i, i) = K1(i, i) = 3/4− 10⁻³ for i∈ {50, 51}

K₁(i, j) = 0 otherwise

and 





K₂(i, i + 1) = K₂(i + 1, i) = 10⁻² for i = 25 K2(i, i) = 3/4− 10⁻² for i∈ {25, 26}

K₂(i, j) = K₁(i, j) otherwise .

Note that each curve has only one sharp transition for d_TV(t) ≤ 1/2. This is consistent with Theorem 1.3. These examples show that multiple sharp transitions may occur for d_TV(t) > 1/2. Note also that the flat part of the curves occupy very large time regions. For instance, the left most curve stays near the value 1/2 for t between 10³and 10⁶.

3. Bounds for mixing time and spectral gap

This section is dedicated to proving Theorems 1.1 and 1.2. In the first two subsections, we treat respectively the upper and lower bounds of the total variation mixing time. This leads to Theorem 1.1. In the third subsection, we provide a

0 5 10 15 20 25 30 35 40 45 50 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 5 10 15 20 25 30 35 40 45 50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1. The curves display the total variation distance of the chains in Remark 2.2, where the left most curve is for K1 and the right most curve is for K2. The curve consists of the points (m, dTV(100^⌊0.1×m⌋)) with m = 1, 2, ..., 50. The right most point of each curve corresponds to dTV(t) with t = 10¹⁰.

relaxation of the choice of i_n in Theorem 1.3. In the last subsection, we introduce a bound on the spectral gap which includes Theorem 1.2.

3.1. An upper bound of the mixing time. Let (Ω, K, π) be an irreducible birth and death chain, where Ω = {0, 1, ..., n} and K has birth rate pi, death rate q_i and holding rate r_i. Let (X_m)^∞_m=0 be a realization of the discrete time chain. Obviously, if N_t is a Poisson process with parameter 1 and independent of (X_m)^∞_m=0, then (X_Nt)_t≥0 is a realization of the continuous time chain. For δ ∈ [0, 1), if (B^m(δ))^∞_m=1 is a sequence of independent Bernoulli(1− δ) trials which are independent of (X_m)^∞_m=0, then Ym^(δ) = X_B(δ)

1 +···+B^(δ)m is a realization of the δ-lazy chain. For 0≤ i ≤ n, we define the first passage time to i by

(3.1) eτi:= inf{t ≥ 0|XN_t = i}, τi^(δ):= min{m ≥ 0|Ym= i},

and simply put τi := τ_i⁽⁰⁾ = min{m ≥ 0|Xm = i}. Briefly, we write Pi(·) for P(·|X0= i) and writeEi, Vari as the expectation and variance underPi. The main result of this subsection is as follows.

Theorem 3.1 (Upper bound). Let (Ω, K, π) be an irreducible birth and death chain with Ω ={0, 1, ..., n}. Let τi = τ_i⁽⁰⁾ be the first passage time to i defined in (3.1).

For ϵ∈ (0, 1) and δ ∈ [1/2, 1),

(3.2) max

{

T_TV^(c)(ϵ), (1− δ)TTV^(δ)(ϵ)

}≤9(E0τ_i0+Enτ_i0)

ϵ² ,

where i0∈ {0, ..., n} satisfies π([0, i0− 1]) ≤ 1/2 and π([i0+ 1, n])≤ 1/2.

Remark 3.1. The authors of [6] obtain a slightly improved upper bound similar to (3.2), which says that

max {

T_TV^(c)(ϵ), (1− δ)TTV^(δ)(ϵ) }≤ (√

ϵ +√

1− ϵ)(E0τ_i0+Enτ_i0)

√ϵ .

Comparing with (3.2), the above inequality has an improved dependence on ϵ.

To understand the right side of (3.2), we introduce the following lemma.

Lemma 3.2. Referring to the setting in (3.1), it holds true that, for i < j, Ei(τ_j^(δ)) =Ei(τ_j)/(1− δ) and Ei(τ_j) =Ei(eτj) =∑j−1

k=iπ([0, k])/(p_kπ(k)).

Proof. The proof is based on the strong Markov property. See [2, Proposition 2]

for a reference on the discrete time case, whereas the continuous time case is an immediate result of the fact{eτi> t} = {τi> N_t}.  Remark 3.2. By Theorem 3.1 and Lemma 3.2, the total variation mixing time for the continuous time and the δ-lazy, with δ ≥ 1/2, discrete time birth and death chain on {0, 1, ..., n} are bounded above by the following term up to a multiple constant.

i∑₀−1 k=0

π([0, k]) pkπ(k) +

∑n k=i0+1

π([k, n]) qkπ(k) ,

where i0∈ {0, ..., n} satisfies π([0, i0− 1]) ≤ 1/2 and π([i0+ 1, n])≤ 1/2.

Remark 3.3. In Theorem 3.1, i0 is unique if π([0, i]) ̸= 1/2 for all 0 ≤ i ≤ n. If π([0, j]) = 1/2, then i0can be j or j + 1, but the right side of (3.2) is the same in either case using Lemma 3.2.

Remark 3.4. Let K be an irreducible birth and death chain with birth, death and holding rates p_i, q_i, r_i and stationary distribution π. Let λ be the spectral gap of K. As a consequence of Lemma 2.1 and theorem 3.1, we obtain, for ϵ∈ (0, 1/2),

λ≥ϵ²log(1/(2ϵ)) 9

(_i

0−1

∑

k=0

π([0, k]) p_kπ(k) +

∑n k=i₀+1

π([k, n]) q_kπ(k)

)−1

,

where i₀is such that π([0, i₀− 1]) ≤ 1/2 and π([i0+ 1, n])≤ 1/2. The maximum of ϵ²log(1/(2ϵ)) on (0, 1/2) is attained at ϵ = 1/(2√

e) and equal to 1/(8e). A similar lower bound of the spectral gap is also derived in [7] with improved constant.

As a simple application of Lemma 3.2, we have Corollary 3.3. Referring to Lemma 3.2, for i≤ j,

Eiτ_j ≤

( 1

π([j, n])− 1 )

Enτ_i. Proof. By Lemma 3.2, one has

Eiτj=

j−1

∑

k=i

π([0, k])

p_kπ(k) , En(τi) =

n−1

∑

k=i

π([k + 1, n]) q_k+1π(k + 1)=

n∑−1 k=i

π([k + 1, n]) p_kπ(k) . The inequality is then given by the fact π([0, k])/π([k + 1, n]) = 1/π([k + 1, n])−1 ≤

1/π([j, n])− 1 for k < j. 

The following proposition is the main technique used to prove Theorem 3.1.

Proposition 3.4. Referring to the setting in (3.1), it holds true that, for j < k, d^(c)_TV(i, t)≤ Pi(max{eτj,eτk} > t) + 1 − π([j, k]),

and

d^(1/2)_TV (i, t)≤ Pi(max{τj^(1/2), τ_k^(1/2)} > t) + 1 − π([j, k]), In particular,

d^(c)_TV(t)≤ E0eτk+Eneτj

t + 1− π([j, k])

and

d^(1/2)_TV (t)≤ 2(E0τ_k^(1/2)+Enτ_j^(1/2))

t + 1− π([j, k]).

In the above proposition, the discrete time case is discussed in Lemma 2.3 in [15]. Our method to prove this proposition is to construct a no-crossing coupling.

We give the proof of the continuous time case for completeness and refer to [15]

for the discrete time case, where a heuristic idea on the construction of no-crossing coupling is proposed.

Proof of Proposition 3.4. Let (Yt)t≥0be another process corresponding to Htwith Y0

= π. Set T := infd {t ≥ 0|Xt = Yt} and Zt := Yt1_{{t≤T }}+ Xt1_{{t>T }}. Clearly, (Xt, Zt)t≥0 is a coupling for the semigroup Htand must be no-crossing according to the continuous time setting. Note that T = inf{t ≥ 0|Xt= Zt} is the coupling time of Xt and Zt. The classical coupling statement implies that

(3.3) d^(c)_TV(i, t)≤ Pi(T > t).

See e.g. [1] for a reference. Note that Xτ_j = j, Xτ_k= k and

Pi(X_eτj ≤ Y_eτj) = π([j, n]), Pi(X_eτk ≥ Y_eτk) = π([0, k]).

As Xt, Ytcan not cross each other without coalescing in advance, this implies Pi(T ≤ max{eτj,eτk}) ≥ Pi(min{eτj,eτk} ≤ T ≤ max{eτj,eτk})

≥ Pi(X_eτj ≤ Yeτ_j, X_eτk≥ Yτe_k)≥ π([j, k]).

Putting this back to (3.3) gives the desired result.

For the last part, note that if i≤ j, then eτj <eτk and, by Markov’s inequality, this implies

Pi(max{eτj,eτk} > t) ≤ P0(eτk> t)≤ E0eτk/t.

Similarly, for i≥ k, one can show that

Pi(max{eτj,eτk} > t) ≤ Pn(eτj> t)≤ Eneτj/t.

For j < i < k, we have

Pi(max{eτj,eτk} > t) ≤ Pi(eτj> t) +Pi(eτk> t)≤Eneτj+E0eτk

t .

 Proof of Theorem 3.1. Set jϵ = min{i ≥ 0|π([0, i]) ≥ ϵ/3} and kϵ = min{i ≥ 0|π([0, i]) ≥ 1 − ϵ/3}. By Proposition 3.4 and Lemma 3.2, the choice of j = jϵ and k = k_ϵ implies that

T_TV^(c)(ϵ)≤ 3(E0τk_ϵ+Enτj_ϵ)

ϵ .

By Corollary 3.3, one has

E0τ_kϵ =E0τ_i0+Ei0τ_kϵ≤ E0τ_i0+ (3

ϵ − 1 )

Enτ_i0 and

Enτ_jϵ=Enτ_i0+Ei0τ_jϵ≤ Enτ_i0+ (3

ϵ − 1 )

E0τ_i0.

Adding up both terms gives the upper bound in continuous time case. The proof for the (1/2)-lazy discrete time case is similar and, by Proposition 3.4, we obtain

T_TV^(1/2)(ϵ) ≤ 18(E0τ_i0+Enτ_i0)/ϵ². For δ ∈ (1/2, 1), note that Kδ = (K_2δ−1)_1/2. Since the birth and death rates of K_2δ−1 are 2(1− δ)pi and 2(1− δ)qi, the above result and Lemma 3.2 lead to T_TV^(δ)(ϵ)≤ 9(E0τ_i0+Enτ_i0)/((1− δ)ϵ²).  3.2. A lower bound of the mixing time. The goal of this subsection is to estab-lish a lower bound on the total variation mixing time for birth and death chains.

Recall the notations in the previous subsection. Let (X_m)^∞_m=0 be an irreducible birth and death chain with transition matrix K and stationary distribution π. Let N_tbe a Poisson process of parameter 1 that is independent of X_m. For 0≤ i ≤ n, let τ_i = min{m ≥ 0|Xm = i} and eτi = inf{t ≥ 0|XNt = i}. Then, the total variation mixing time satisfies

(3.4) d_TV(0, t)≥ K^t(0, [0, i− 1]) − π([0, i − 1]) ≥ P0(τ_i> t)− π([0, i − 1]) and

(3.5) d^(c)_TV(0, t)≥ Ht(0, [0, i− 1]) − π([0, i − 1]) ≥ P0(eτi> t)− π([0, i − 1]).

Brown and Shao discuss the distribution ofeτi in [3], of which proof also works for the discrete time case. In detail, if −1 < β1<· · · < βi < 1 are the eigenvalues of the submatrix of K indexed by{0, ..., i − 1} and λj = 1− βj, then

(3.6) P0(τi> t) =

∑i j=1



∏

k̸=j

λk

λk− λj



 (1 − λ^j)^t

and

(3.7) P0(eτi> t) =

∑i j=1



∏

k̸=j

λ_k λk− λj



 e^−tλj.

Note that, under P0, eτi is the sum of independent exponential random variables with parameters λ₁, ..., λ_i. If β₁ > 0, then τ is the sum of independent geometric random variables with parameters λ1, ..., λi. In discrete time case, the requirement β1 > 0 holds automatically for the δ-lazy chain with δ≥ 1/2. The above formula leads to the following theorem.

Theorem 3.5 (Lower bound). Let K be the transition matrix of an irreducible birth and death chain on {0, 1, ..., n}. Let τi = τ_i⁽⁰⁾ be the first passage time to i defined in (3.1). For δ∈ [1/2, 1),

min{TTV^(c)(1/10), 2(1− δ)TTV^(δ)(1/20)} ≥ max{E0τi₀,Enτi₀}

6 ,

where i₀∈ {0, ..., n} satisfies π([0, i0− 1]) ≤ 1/2 and π([i0+ 1, n])≤ 1/2.

Proof of Theorem 3.5. First, we consider the continuous time case. Let λ1, ..., λi

be eigenvalues of the submatrix of I− K indexed by 0, ..., i − 1 and eτi,1, ...,eτi,i be independent exponential random variables with parameters λ1, ..., λi. By (3.7), eτi

andeτi,1+· · · + eτi,iare identically distributed underP0and, by (3.5), this implies d^(c)_TV(0, t)≥ P(eτi,1+· · · + eτi,i> t)− π([0, i − 1]).

It is easy to see that E0eτi= 1

λ₁ +· · · + 1

λ_i, Var0(eτi) = 1

λ²₁ +· · · + 1 λ²_i.

Let a ∈ (0, 1) and consider the following two cases. If 1/λj > aE0eτi for some

For the discrete time case, note that the eigenvalues of the submatrix of I− K_1/2 = ¹₂(I − K) indexed by 0, ..., i − 1 are λ1/2, ..., λi/2. Let τi,1, ..., τi,i be independent geometric random variables with success probabilities λ1/2, ..., λi/2.

Replacing K with K_1/2in (3.4), we obtain

d^(1/2)_TV (0, t)≥ P0(τ_i,1+· · · + τi,i> t)− π([0, i − 1]). where the first inequality use the fact that s log(1− 3/(2s)) is increasing on [2, ∞).

Hence, we have T_TV^(1/2)(0, 1/20) ≥ E0τ_i^(1/2)

0 /12 = E0τi₀/6. For δ > 1/2, the com-bination of the above result and the observation K_δ = (K_2δ−1)_1/2 implies that T_TV^(δ)(0, 1/20)≥ E0τi₀/(12(1− δ)).

The analysis from the other end point gives the other lower bound. This finishes

the proof. 

3.3. Relaxation of the median condition. In some cases, it is not easy to determine the value of in in Theorem 1.3. Let tn be the constants in Theorem 3.1.

For c ∈ (0, 1), let in(c) ∈ {0, ..., n} be the state such that πn([0, in(c)− 1]) ≤ c, π_n([i_n(c) + 1, n])≤ 1 − c and let tn(c) be the following constant

tn(c) =

in∑(c)−1 k=0

πn([0, k]) πn(k)pn,k

+

∑n k=i_n(c)+1

πn([k, n]) πn(k)qn,k

.

Assume that c≥ 1/2. In this case, if in is the smallest median, then in≤ in(c) and

i_n∑(c)−1 k=i_n

π([0, k]) π_n(k)p_n,k =

i∑_n(c) k=i_n+1

πn([0, k− 1]) π_n(k)q_n,k . Note that, for i_n< k≤ in(c),

1

2 ≤ πn([0, i_n])≤ π_n([0, k− 1])

πn([k, n]) ≤ 1

πn([in(c), n])≤ 1 1− c.

This implies tn/2≤ tn(c)≤ tn/(1− c). Similarly, for c ≤ 1/2, one can show that tn/2≤ tn(c)≤ tn/c. Combining both cases gives

(3.9) t_n/2≤ tn(c)≤ tn/ min{c, 1 − c}.

As a consequence of the above discussion, we obtain the following theorem.

Theorem 3.6. Referring to Theorem 1.3. For n≥ 1, let jn∈ {0, 1, ..., n} and set

t^′_n = max





j∑n−1 k=0

πn([0, k]) π_n(k)p_n,k,

∑n k=j_n+1

πn([k, n]) π_n(k)q_n,k



.

Suppose that

0 < lim inf

n→∞ π_n([0, j_n])≤ lim sup

n→∞ π_n([0, j_n]) < 1.

Then, Theorem 1.3 remains true if tn is replaced by t^′_n.

Proof. The proof comes immediately from (3.9) with c = πn([0, jn]).  We use this observation to bound the cutoff time in the following theorem.

Theorem 3.7. Referring to Theorem 1.3. Suppose that Fc has a total variation cutoff. Then, for any ϵ∈ (0, 1),

2 log 2

5 ≤ lim inf

n→∞

T_n,^(c)_TV(ϵ)

tn ≤ lim sup

n→∞

T_n,^(c)_TV(ϵ) tn ≤ 2

Proof of Theorem 3.7. The upper bound is given by Remark 3.1 and the fact, max{s, t} ≥ (s + t)/2, whereas the lower bound is obtained by applying a = 2/5

and b = a log(2/(1 + 2ϵ)) in (3.8) with ϵ→ 0. 

3.4. Bounding the spectral gap. This subsection is devoted to poviding bounds on the specral gap for birth and death chains. As the graph associated with a birth and death chain is a path, weighted Hardy’s inequality can be used to bound the spectral gap. We refer to the Appendix for a detailed discussion of the following results. See Theorems A.1-A.3.

Theorem 3.8. Consider an irreducible birth and death chain on {0, ..., n} with birth, death and holding rates p_i, q_i, r_i and stationary distribution π. Let λ be the spectral gap and set, for 0≤ i ≤ n,

C(i) = max



max

j:j<i i−1

∑

k=j

π([0, j]) π(k)pk

, max

j:j>i

∑j k=i+1

π([j, n]) π(k)qk



. Then, for 0≤ m ≤ n,

1

4C(m) ≤ λ ≤ 1

min{π([0, m]), π([m, n])}C(m).

In particular, if M is a median of π, that is, π([0, M ])≥ 1/2 and π([M, n]) ≥ 1/2, then

1

4C(M ) ≤ λ ≤ 2 C(M ).

Theorem 3.9. Consider an irreducible birth and death chain on {0, ..., n} with birth, death and holding rates pi, qi, ri and stationary distribution π. Let λ be the spectral gap and set N =⌈n/2⌉. Suppose that pi= qn−i for 0≤ i ≤ n. Then,

1

4C ≤ λ ≤ 1 C, where

C = max

0≤i≤N−1



π([0, i])

N∑−1 j=i

1 π(j)pj



 if n is even, and

C = max

0≤i≤N−1



π([0, i])



^N∑⁻²

j=i

1 π(j)pj

+ 1

2π(N − 1)pN−1







 if n is odd.

Remark 3.5. In [18], the author also obtained bounds similar to Theorem 3.9 for the case π(i) ≥ π(i + 1) with 0 ≤ i < n/2 using the path technique. For more information on path techniques, see [11, 12, 14] and the references therein.

4. Examples

In this section, we will apply the theory developed in the previous section to examples of special interest. First, we give a criterion on the cutoff using the birth and death rates.

Theorem 4.1 (Cutoffs from birth and death rates). Let F = {(Ωn, K_n, π_n)|n = 1, 2, ...} be a family of irreducible birth and death chains on Ωn ={0, 1, ..., n} with

birth rate, pn,i, death rate qn,i and holding rate rn,i. Let λn be the spectral gap of

Furthermore, the following are equivalent.

(1) Fc has a cutoff in total variation.

(2) For δ∈ (0, 1), Fδ has a cutoff in total variation.

(3) Fc has precutoff in total variation.

(4) For δ∈ (0, 1), Fδ has a precutoff in total variation.

(5) t_n/ℓ_n→ ∞.

The above theorem is obvious from Theorems 2.2, 3.6 and 3.8. We use two classical examples, simple random walks and Ehrenfest chains, to illustrate how to apply Theorem 4.1 to determine the total variation cutoff and mixing times.

Example 4.1 (Simple random walks on finite paths). For n≥ 1, the simple random walk on{0, ..., n} is a birth and death chain with pn,i= qn,i+1= 1/2 for 0≤ i < n and rn,0= rn,n = 1/2. It is clear that Knis irreducible and aperiodic with uniform stationary distribution. Let tn, ℓn be the constants in Theorem 4.1. It is an easy exercise to show that ℓn≍ n²≍ tn. By Theorem 4.1, neither Fc norFδ has total variation precutoff, but T_n,^(c)_TV(ϵ)≍ n²≍ Tn,^(δ)TV(ϵ) for ϵ∈ (0, 1/2) and δ ∈ (0, 1). In fact, one may use a hitting time statement to prove that the mixing time has order at least n², when ϵ∈ [1/2, 1). This implies that the above approximation of mixing time holds for ϵ∈ (0, 1).

Example 4.2 (Ehrenfest chains). Consider the Ehrenfest chain on{0, ..., n}, which is a birth and death chain with rates pn,i = 1− i/n and qn,i = i/n. It is obvious that K_n is irreducible and periodic with stationary distribution π_n(i) = 2⁻ⁿ(_n

i

). An application of the representation theory shows that, for 0≤ i ≤ n, 2i/n is an eigenvalue of I−Kn. Let λ_n, s_nbe the constants in Theorem 2.2. Clearly, λ_n= 2/n and sn≍ n log n and, by Theorem 2.2, both FcandFδ have a total variation cutoff.

Note that, as a simple corollary, one obtains the non-trivial estimates

⌈ⁿ∑2⌉−1

For a detailed computation on the total variation and the L²-distance, see e.g. [9].

In the next subsections, we consider birth and death chains of special types.

4.1. Chains with valley stationary distributions. In this subsection, we con-sider birth and death chains with valley stationary distribution. For n ≥ 1, let Ωn={0, 1, ..., n} and Kn be an irreducible birth and death chain on Ωnwith birth, One can derive a similar inequality from the other end point and this yields

ℓ_n≥ 1

The following theorem is an immediate consequence of the above discussion and Theorem 4.1.

Theorem 4.2. Let F = {(Ωn, Kn, πn)|n = 1, 2, ...} be a family of birth and death For an illustration of the above theorem, we consider the following Markov chains.

For n ≥ 1, let Ωn ={0, 1, ..., n}, πn be a non-uniform probability distribution on Ωn satisfying (4.1) and Mn be a transition matrix given by

(4.2) Mn(i, j) =

Note that Mn is the Metropolis chain for πn associated to the simple random walk on Ωn. For more information on the Metropolis chain, see [8] and the references therein. The next theorem is a corollary of Theorem 4.2.

Theorem 4.3. Let F = {(Ωn, Mn, πn)|n = 1, 2, ..} be the family of Metropolis

where ˇcn,a, ˆcn,a are normalizing constants. Let ˇF, ˆF be families of the Metropolis chains for ˇπn,a, ˆπn,a associated to the simple random walks on{0, ±1, ..., ±n}, that

Let ˇλn,a, ˆλn,aand ˇTn,a, ˆTn,abe the spectral gaps and total variation mixing times

The above result in continuous time case is also obtained in [18].

To see the cutoff for ˆF, let

By Theorems 3.1-3.5, we have 2t_n We collect the above results in the following theorem.

Theorem 4.4. For n≥ 1, let an> 0 and ˇπn,an, ˆπn,an be probability measures given

4.2. Chains with monotonic stationary distributions. In this subsection, we consider birth and death chains with monotonic stationary distributions. For n≥ 1, let Ωn ={0, 1, ..., n} and Kn be a birth and death chain on Ωn with birth, death

Using a discussion similar to that in front of Theorem 4.2, one can show that

t_n≍ max This leads to the following theorem.

Theorem 4.5. Let F = {(Ωn, Kn, πn)|n = 1, 2, ...} be a family of irreducible birth and death chains with Ω_n = {0, 1, ..., n} and birth, death and holding rates p_n,i, q_n,i, r_n,i. Let λ_n, T_n,TV be the spectral gap and total variation mixing time of Moreover,Fc and Fδ have a total variation cutoff if and only if

un/vn→ ∞, (unpn,jn)/(wnpn,1)→ ∞.

This implies By Theorem 4.5,Fc andFδ have a total variation cutoff and

λn ≍ n^2βn−2, T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ n^2−βn, ∀ϵ ∈ (0, 1/2), δ ∈ (0, 1).

Set jn= n[1− (log n)^1−βn]. The above computation leads to

π_n([0, j_n])≍ πn([j_n, n]), u_n≍ n²(log n)^1−βn, v_n≍ n²(log n)^2−2βn≍ wn. By Theorem 4.5, bothFc andFδ have a total variation cutoff and, for ϵ∈ (0, 1/2) and δ∈ (0, 1),

λn≍ n⁻²(log n)^2βn−2, T_n,^(c)_TV(ϵ)≍ n²(log n)^1−βn≍ Tn,^(δ)TV(ϵ).

Case 4: fn(x) = exp{αn[log(x + 1)]^βn} with supnαn <∞ and supnβn ≤ 1.

Observe that, for α > 0 and β≤ 1, d

dx (

(x + 1)eα[log(x+1)]^β)

=(

1 + αβ[log(x + 1)]^β−1)

eα[log(x+1)]^β. This implies that, uniformly for n/4≤ i < m ≤ n,

F_n(i)≍ (i + 1)e^{αn[log(i+1)]βn}, G_n(i, m)≍

( i + 1

e^{αn[log(i+1)]βn} − m + 1 e^{αn[log(m+1)]βn}

) . Letting jn=⌊n/2⌋ implies

π_n([0, j_n])≍ πn([j_n, n]), u_n≍ vn≍ wn≍ n². By Theorem 4.5, we have

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ λ⁻¹n ≍ n², ∀ϵ ∈ (0, 1/2), δ ∈ (0, 1), and there is no total variation cutoff forFc orFδ.

4.3. Chains with symmetric stationary distributions. This subsection is ded-icated to the study of birth and death chains with symmetric stationary distribu-tions. Let K be an irreducible birth and death chain on{0, ..., n} with stationary distribution π. Note that π is symmetric at n/2, that is, π(n− i) = π(i) for 0≤ i ≤ n/2, if and only if

pipn−i−1= qi+1qn−i, ∀0 ≤ i ≤ n/2.

By the symmetry of π, we will fix jn=⌊n/2⌋ when applying Theorem 4.1.

Consider a family of irreducible birth and death chains,F = {(Ωn, Kn, πn)|n = 1, 2, ...} with Ωn ={0, 1, ..., n}. Let pn,i, qn,i, rn,i be respectively the birth, death and holding rates of Kn and tn, ℓn be constants in Theorem 4.1. Assume that πn

is symmetric at n/2. Continuously using the fact (a + b)/2≤ max{a, b} ≤ a + b for a≥ 0, b ≥ 0, we obtain

t_n ≍ ∑

k:k≤n/2

π_n([0, k]) πn(k) min{pn,k, qn,n−k} and

ℓ_n ≍ max

j:j≤n/2

∑

k:j≤k≤n/2

π_n([0, j])

πn(k) min{pn,k, qn,n−k}. Theorem 4.1 can be rewritten as follows.

Theorem 4.6. LetF = {(Ωn, Kn, πn)|n = 1, 2, ...} be a family of irreducible birth and death chains with Ωn = {0, 1, ..., n}. Let λn and pn,i, qn,i, rn,i be the spectral gap and the birth, death and holding rates of Kn. Assume that

p_n,ip_n,n−i−1= q_n,i+1q_n,n−i, ∀0 ≤ i ≤ n/2.

Then, for ϵ∈ (0, 1/2) and δ ∈ (0, 1),

λ_n≍ 1/ℓn, T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ tn, where

tn = ∑

k:k≤n/2

πn([0, k]) π_n(k) min{pn,k, q_n,n−k} and

ℓn= max

j:j≤n/2



πn([0, j]) ∑

k:j≤k≤n/2

1

πn(k) min{pn,k, qn,n−k}



. Moreover, the following are equivalent.

(1) Fc has a cutoff in total variation.

(2) For δ∈ (0, 1), Fδ has a cutoff in total variation.

(3) Fc has a precutoff in total variation.

(4) For δ∈ (0, 1), Fδ has a precutoff in total variation.

(5) tn/ℓn→ ∞.

The next theorem considers a perturbation of birth and death chains which has the same stationary distribution as the original chains. The new chains keep the order of mixing time and spectral gap unchanged.

Theorem 4.7. Consider the family in Theorem 4.6 and assume that pn,ipn,n−i−1= qn,i+1qn,n−i, ∀0 ≤ i ≤ n/2.

For n ≥ 1, let An ⊂ {0, ..., n − 1}, cn,i ∈ [0, 1] for i ∈ An and eKn be a birth and death chain on Ωn with birth and death rates, epn,i,eqn,i, satisfying







epn,i= c_n,ip_n,i+ (1− cn,i) min{pn,i, q_n,n−i} for i ∈ An, eqn,i+1= qn,i+1epn,i/pn,i for i∈ An, epn,i= p_n,i, eqn,i+1= q_n,i+1 for i /∈ An.

Let λn, eλn and Tn,TV(ϵ), eTn,TV(ϵ) be the spectral gaps and total variation mixing times of Kn, eKn. Then, given ϵ∈ (0, 1/2) and δ ∈ (0, 1),

eλn≍ λn, Te_n,^(c)_TV(ϵ)≍ Tn,^(c)TV(ϵ)≍ eT_n,^(δ)_TV(ϵ)≍ Tn,^(δ)TV(ϵ), where the approximation is uniform on the choice of A_n, c_n,i.

Proof. The approximation of the spectral gap and the total variation mixing time is immediate from Theorem 4.6, whereas the uniformity of the approximation is

given by Theorems 3.1, 3.5 and 3.8. 

Example 4.4. For n≥ 1, let Kn be a birth and death chain on {0, 1, ..., 2n} given by

Kn(i, i + 1) = Kn(i + 1, i) = {

1/2 for even i 1/(2n) for odd i .

By Theorem 4.7, the mixing time and spectral gap of Knare comparable with those of eKn, where eKn(i, i + 1) = eKn(i + 1, i) = 1/(2n) for 0 ≤ i < 2n. Let F be the family consisting of K_n. By Theorem 4.6, neitherFc norFδ has a total variation precutoff and T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ) ≍ λ⁻¹n ≍ n³ for all ϵ ∈ (0, 1/2) and δ ∈ (0, 1), which is nontrivial.

Next, we consider simple random walks on finite paths with bottlenecks. For n≥ 1, let kn ≤ n and xn,1, ..., xn,k_n be positive integers satisfying 1≤ xn,i< xn,i+1≤ n for i = 1, ..., kn− 1. Let Kn be the birth and death chain on{0, 1, ..., n} of which birth, death and holding rates are given by

(4.6) p_n,i−1= q_n,i= {

1/2 for i /∈ {xn,1, ..., x_n,kn} ϵn,j for i = xn,j, 1≤ j ≤ kn

,

where ϵ_n,j ∈ (0, 1/2] for 1 ≤ j ≤ kn. Clearly, K_n is irreducible and the stationary distribution, say π_n, is uniform on{0, 1, ..., n}. The following theorem is immediate from Theorems 4.6.

Theorem 4.8. LetF be a family of birth and death chains given by (4.6) and λn

be the spectral gap of K_n. For n≥ 1, set tn= n²+

kn

∑

i=1

min{xn,i, n + 1− xn,i} ϵn,i

and

ℓ_n= n²+ max

j:j≤n/2





∑

i:|xn,i−n/2|≤j

n/2 + 1− j ϵn,i



. Then, for all ϵ∈ (0, 1/2) and δ ∈ (0, 1),

T_n,^(c)_TV(ϵ)≍ Tn,^(δ)TV(ϵ)≍ tn, λn≍ 1/ℓn. Furthermore, the following are equivalent.

(1) Fc has a cutoff in total variation.

(2) For δ∈ (0, 1), Fδ has a cutoff in total variation.

(3) Fc has precutoff in total variation.

(4) For δ∈ (0, 1), Fδ has a precutoff in total variation.

(5) t_n/ℓ_n→ ∞.

Remark 4.1. Let t_n, ℓ_n be the constants in Theorem 4.8. Then, tn≍ n²+ ∑

j∈Ln

xn,j

ϵ_n,j + ∑

j∈Rn

n + 1− xn,j

ϵ_n,j and

ℓn≍ n²+ max

i∈Ln

∑

j∈Ln:j≥i

xn,i

ϵ_n,j + max

i∈Rn

∑

j∈Rn:j≤i

n + 1− xn,i

ϵ_n,j . where Ln ={i : xn,i≤ n/2} and Rn={i : xn,i> n/2}.

Theorem 1.4 considers a special case of Theorem 4.8 with ϵn,i= ϵnfor 1≤ i ≤ kn. It is clear from Theorem 1.4 that if k_n is bounded, then no cutoff exists for Fc or Fδ. The following example shows a case of cutoffs for the family in Theorem 1.4.

Example 4.5. LetF be the family in Theorem 1.4, with kn =⌊n^1/3⌋ − 1 and xn,i=

⌊ n^5/6 n^1/3− i

⌋

, ∀1 ≤ i ≤ kn.

Clearly, for n large enough, xn,i̸= xn,j when i̸= j. Let an, bn be the constant in Theorem 1.4. It is not hard to show that

Clearly, for n large enough, xn,i̸= xn,j when i̸= j. Let an, bn be the constant in Theorem 1.4. It is not hard to show that

在文檔中 馬可夫鏈的熵收斂 (頁 39-64)