Local large deviations and the strong renewal theoremstaff.matapp.unimib.it/~fcaraven/download/slides/moscow-SPA-201… · SRT for Renewal ProcessesSRT for Random WalksLocal Large

SRT for Renewal Processes SRT for Random Walks Local Large Deviations Proof of the SRT

Local large deviationsand the strong renewal theorem

Francesco Caravenna

Universita degli Studi di Milano-Bicocca

Moscow, SPA 2017

Francesco Caravenna The strong renewal theorem 26 July 2017 1 / 27


Joint work with Ron Doney

(Picture taken from MFO website)


https://owpdb.mfo.de/detail?photo_id=15417


Overview

Necessary and sufficient conditions

for the Strong Renewal Theorem (SRT)

Problem dates back to Garsia and Lamperti (1962)

First focus on renewal processes (random walks with positive increments)

then on general random walks (much harder!)

Key tool: local large deviation estimate of independent interest

For simplicity we stick to the lattice case, but everything applies tonon-lattice distributions



Outline

1. SRT for Renewal Processes

2. SRT for Random Walks

3. Local Large Deviations

4. Proof of the SRT



Outline




4. Proof of the SRT



The classical renewal theorem

Random walk Sk := X1 + X2 + . . .Xk with positive increments

(Xi ) i.i.d. Xi ∈ N = {1, 2, . . .} aperiodic

0 S1 S2 S3 ... 0 S1 S2 S3 ... n

Renewal (Green) function u(n) := P(S visits n) =∑k≥0

P(Sk = n)

Renewal Theorem (Erdos, Feller, Pollard 1949)

limn→∞

u(n) =1

E[X ]= 0 if E[X ] =∞



The case of infinite mean

When E[X ] =∞, at which rate u(n)→ 0 ?

Define the truncated mean

m(n) := E[X ∧ n] = E[min{X , n}]

Guess (SRT) u(n) ∼n→∞

c

m(n)for some c ∈ (0,∞)

Assumptions are needed! But there is hope

Lemma (Erickson 1973)

n

m(n)≤

n∑i=1

u(i) ≤ 2n

m(n)



Asymptotically stable renewal processes

Tail Assumption

P(X > n) ∼n→∞

L(n)

nαα ∈ (0, 1) L(·) slowly varying

Explicit computation m(n) := (1− α) E[X ∧ n] ∼n→∞

L(n) n1−α

Theorem (Garsia, Lamperti 1962) (Erickson 1970)

lim infn→∞

u(n)

1/m(n)= cα :=

sinπα

π

I If α > 12 lim inf can be replaced by lim (the SRT holds)

I If α ≤ 12 there are counterexamples!



Sufficient conditions for the SRT

Local Assumption

P(X = n) ≤ CL(n)

n1+α

Theorem (Doney 1997) (Williamson 1968) (Vatutin, Topchii 2013)

Under Local Assumption, the SRT holds also for α ≤ 12

u(n) = P(S visits n) ∼n→∞

cαm(n)

=cα

L(n) n1−α

Sufficient conditions weaker than the Local Assumption (integral criteria)were given more recently in (Chi 2013, 2015)



What can go wrong?

Without Local Assumption, we only know that

P(X = n) = o(P(X ≥ n)

)= o(1)

L(n)

nα

where o(1) can be as slow as we wish (along subsequences)

P(S visits n) ≥ P(X1 = n) can be � 1

L(n) n1−αif α ≤ 1

2

The SRT fails because a single step (more generally, few steps)

can give “atypical” contribution to P(S visits n)

We must find optimal conditions on X to rule this out



Necessary and sufficient conditions

Define a weighted sum of P(X = n), P(X = n − 1), . . . , P(X = n − δn)

I+(δ; n) :=δn∑r=1

wr P(X = n − r) wr :=r2α−1

L(r)2

Definition

We say that I+(δ; n) is asymptotically negligible (a.n.) iff

I+(δ; n) � 1

m(n)as n→∞ , δ → 0

More precisely: limδ→0

lim supn→∞

I+(δ; n)

1/m(n)= 0



SRT for renewal processes

Recall the Tail Assumption

P(X > n) ∼n→∞

L(n)

nαα ∈ (0, 1) L(·) slowly varying

Theorem (SRT for renewal processes) (Caravenna, Doney 2016)

I If α > 12 the SRT holds with no further assumption

I If α ≤ 12 the SRT holds if and only if I+(δ; n) is a.n.

The weighted sum I+(δ; n) is explicit, but slightly involved

We give a practical sufficient condition in terms of probability of intervals



A nearly optimal condition

Corollary

For α ≤ 12 , a sufficient condition for the SRT is

P(n < X ≤ n + r) ≤ CL(n)

nα

( rn

)γ∀r = 1, . . . , n (?)

for some C <∞ and γ > 1− 2α

P(X ≤ n + r |X > n) ≤ C( rn

)γ(?)

Condition (?) is actually nearly optimal

Corollary

A necessary condition for the SRT is that (?) holds for every γ < 1− 2α



Outline




4. Proof of the SRT



General random walks

Random walk Sk := X1 + X2 + . . .Xk on Z

(Xi ) i.i.d. Xi ∈ Z aperiodic

Tail Assumption

P(X > n) ∼n→∞

pL(n)

nαP(X < −n) ∼

n→∞qL(n)

nα

p > 0, q ≥ 0 α ∈ (0, 1) L(·) slowly varying

We use the shorthandm(n) := L(n) n1−α

which is ' E[|X | ∧ n]



A first renewal theorem

Renewal (Green) function u(n) :=∑k≥0

P(Sk = n)

Theorem (Williamson 1968) (Erickson 1971)

lim infn→∞

u(n)

1/m(n)= cα,p,q

I If α > 12 lim inf can be replaced by lim (the SRT holds)

I If α ≤ 12 there are counterexamples

The Local Assumption is again sufficient condition for the SRT

P(X = n) ≤ CL(n)

|n|1+α



A first condition

Extend the previously introduced weighted sum (note the |r |)

I1(δ; n) :=δn∑|r |=1

wr P(X = n + r) wr :=|r |2α−1

L(|r |)2

I1(δ; n) depends only on X+, but SRT may also depend on X−

Proposition

For random walks, a sufficient (not necessary) condition for the SRT is

I1(δ; n) and I1(δ;−n) are both a.n.

In particular, a sufficient condition is that (?) holds for both n and −n

In general, assuming that I1(δ;−n) is a.n. is too strong



SRT for random walks

For k ≥ 2 define a k-tuple weighted sum, depending on δ, η ∈ (0, 1)

Ik(δ, η; n) :=∑|r1|≤δn

|rj |≤η|rj−1| for 2≤j≤r

P(X = n + r1) P(X = −r1 + r2) · · ·

· P(X = −rk−1 + rk)|rk |(k+1)α−1

L(|rk |)k+1

Theorem (SRT for random walks) (Caravenna, Doney 2016)

Consider a random walk satisfying the Tail Assumption with p, q > 0

I If α > 12 the SRT holds with no further assumption

I If 13 < α < 1

2 the SRT holds if and only if I1(δ; n) is a.n.

I Given k ≥ 2, if 1k+2 < α < 1

k+1 the SRT holds if and only if

Ik(δ, η; n) is a.n., for every fixed η ∈ (0, 1)



Further considerations

I Necessary and sufficient conditions available also for α = 1k , k ∈ N,

in terms of slightly modified quantities Ik(δ, η; n)

I Given α ∈ (0, 1) let k ∈ N be such that 1k+2 < α < 1

k+1

SRT holds ⇐⇒ Ik is a.n. ⇐⇒ Ik′ is a.n. ∀k ′ ∈ N

I There are examples where I1(δ; x) is a.n. but I2(δ, η; x) is not a.n.

I When q = 0 the previous conditions are sufficient, but we don’texpect them to be necessary



Outline




4. Proof of the SRT



Asymptotically stable random walks

Consider a random walk Sk = X1 + . . .+ Xk on Z

P(X > n) ∼n→∞

pL(n)

nαP(X < −n) ∼

n→∞qL(n)

nα

We allow both α ∈ (0, 1) and α ∈ (1, 2) (with E[X ] = 0)

We know thatSkak

d−−−−−→k→∞

Y α-stable law(ak ∼ L(k) k

1α

)Gnedenko LLT

P(Sk = n) =1

ak

(ϕ( n

ak

)+ o(1)

)=

o(1)

akfor n� ak

We improve on this bound, giving a quantitative estimate of o(1)



Local large deviations

Theorem (Local Large Deviations) (Caravenna, Doney 2016)

P(Sk = n) ≤ C

ak

k L(n)

nα

For every γ ∈ (0, 1]

P(Sk = n, max{X1, . . . ,Xk} ≤ γn

)≤ Cγ

ak

(k L(n)

nα

)d 1γ e

(�)

I Improves on Gnedenko’s bound with no extra assumptions

I Key tool for the SRT

I (�) is a local version of the Fuk-Nagaev inequality

I Extended to the Cauchy case α = 1 in (Berger 2017)



Outline




4. Proof of the SRT



Probabilistic proof: strategy

Goal: SRT

u(n) ∼n→∞

cαm(n)

=cα

L(n) n1−α

u(n) =∑k≥0

P(Sk = n) =∑

k≤δ nα

L(n)︸︷︷︸(1)

+∑

δ nα

L(n)<k≤ 1δ

nα

L(n)︸︷︷︸(2)

+∑

k> 1δ

nα

L(n)︸︷︷︸(3)

I (3) is always negligible by Gnedenko’s LLT P(Sk = n) ≤ Cak

I (2) is a Riemann sum approximation: by P(Sk = n) ∼ 1akϕ( n

ak)

(. . . ) it gives full contribution to SRT as n→∞, δ → 0

SRT holds ⇐⇒ (1) � 1m(n) is asymptotically negligible



Necessity: SRT ⇒ I+(δ; n) is a.n.

P(Sk = n) ≥ P(Sk = n, ∃!Xi > Cak)

= (. . .)

≥ c∑

1≤r≤δn

P(X = n − r)k

ak1{ rα

L(r)<k≤2 rα

L(r)}

Then (1) =∑

k≤δ nα

L(n)

P(Sk = n)

≥ c∑

1≤r≤δn

P(X = n − r)∑

rα

L(r)<k≤2 rα

L(r)

k

ak

≥ c∑

1≤r≤δn

P(X = n − r)r2α−1

L(r)2= I+(δ; n)



Sufficiency: I+(δ; n) is a.n. ⇒ SRT

Fix suitable γ ∈ (0, 1). Using local large deviation

P(Sk = n, max{X1, . . . ,Xk} ≤ γn) is negligible

P(Sk = n, max{X1, . . . ,Xk} > γn) ≤ k∑

r≤(1−γ)n

P(X = n−r) P(Sk−1 = r)

Then (1) ≤∑

r≤(1−γ)n

P(X = n − r)∑

k≤δ nα

L(n)

k P(Sk−1 = r)

I We can replace n by r in the inner sum, using that I+(δ; n) is a.n.

I Then the inner sum becomes (1) with an extra factor k

I This helps! (Backward) induction argument allows to conclude



Thanks


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Local large deviations and the strong renewal theoremstaff.matapp.unimib.it/~fcaraven/download/slides/moscow-SPA-201… · SRT for Renewal ProcessesSRT for Random WalksLocal Large