random walks applied probability and queues by s....

Basic definitionsLadder processes

ClassificationAppendix

Random WalksApplied probability and queues by S. Asmussen

Lenka Slámová[email protected]

Stochastic modelling in economics and financeNovember 19th, 2012

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1 Basic definitions

2 Ladder processes

3 Classification

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1 Basic definitions

2 Ladder processes

3 Classification

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Basic definitionsConsider a random walk Sn = X1 + · · ·+ Xn, S0 = 0.Xk are i.i.d. with common distribution F

Terminology and notation

Mn is the partial maximum max0≤k≤n SkM is the total maximum sup0≤k 0} is the first (strict) ascending ladder

epoch or the entrance time to (0,∞)Sτ+ is the first (strict) ascending ladder height (defined on{τ+



Basic definitions - continued

Descending ladder

τ− = τw− = inf{n ≥ 1 : Sn ≤ 0} is the first (weak) descending ladder

epoch or the entrance time to (−∞, 0]Sτ− is the first (weak) descending ladder height (defined on{τ−



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Notes

There is a clear asymmetry in the definitions of τ+ and τ− (strictinequality Sn > 0 vs. weak inequality Sn ≤ 0).Weak ascending and strict descending ladder epochs can be definedin an obvious way by

τw+ = inf{n ≥ 1 : Sn ≥ 0}, τ s− = inf{n ≥ 1 : Sn < 0}.

Sτw+ , Sτ s− , Gs+, Gw− may be needed if we want to study minimum

inf0≤k



Useful lemmas

Proposition 1.1.Define ζ = P(Sτ− = 0) = P(τ− 6= τ s−). Then ζ < 1 andζ = P(Sτw+ = 0) = P(τ+ 6= τ

w+ ) and if δ0 is the distribution degenerate at

0, then

Gw+ = ζδ0 + (1− ζ)G+, G− = ζδ0 + (1− ζ)G s−.

The Proof of this Proposition is based on simple observation

Lemma 1.2.Let n be fixed and define S∗k = Sn − Sn−k = Xn−k+1 + · · ·+ Xn,k = 0, . . . , n. Then

{S∗k }n0d= {Sk}n0.

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Proof.For any n we get by Lemma 1.2.

P(Sτw+ = 0, τw+ = n) = P(S∗0 = 0,S∗k < 0, k = 1, . . . , n − 1,S∗n = 0) =

= P(S0 = 0,Sn − Sn−k < 0, k = 1, . . . , n − 1,Sn = 0) == P(S0 = 0,Sl > 0, l = 1, . . . , n − 1,Sn = 0) = P(Sτ− = 0, τ− = n)

If we sum over n we obtain ζ = P(Sτ− = 0) = P(Sτw+ = 0). Finally1− ζ = P(Sτ− < 0) ≥ P(X1 < 0) > 0.

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1 Basic definitions

2 Ladder processes

3 Classification

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Ladder processes

By iterating definitions of τ+ and τ− we define sequences{τ+(n)}, {τ−(n)}.

Ladder process

Let τ+(1) = τ+ and τ−(1) = τ−τ+(n + 1) = inf{k > τ+(n) : Sk > Sτ+(n)}τ−(n + 1) = inf{k > τ−(n) : Sk ≤ Sτ−(n)}

The points of the plane (τ+(n),Sτ+(n)) are called ascending ladderpoints(τ+(n)} is called ascending ladder epoch process{Sτ+(n)} is called ascending ladder process

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Illustration

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Connection to renewal processes

Segments of random walk between ascending (or descending) ladderpoints are i.i.d. replicates.Ascending ladder epoch process {τ+(n)} is a discrete time renewalprocess with governing probabilities fn = P(τ+ = n), terminating ifand only if ||G+|| < 1.The ascending ladder height process {Sτ+(n)} is a renewal processgoverned by G+, proper if and only if ||G+|| = 1The overshoot process {B(u)}u≥0 of the random walk coincides withthe forward recurrence time process.

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Theorem 2.1.The overshoot B(u) is proper if and only if the ascending ladder heightprocess is nonterminating, i.e. ||G+|| = 1. In that case, it holds as u →∞that B(u) d→∞ if ESτ+ =∞, whereas if ESτ+



Distribution of the maximum

Clearly the maximum M = sup{Sτ+(n) : τ+(n)



Theorem 2.3.

(a) P(Sn > Sk , k = 0, . . . , n − 1,Sn ∈ A) = P(Sk > 0, k = 1, . . . , n,Sn ∈A)

(b) U+(A) = E∑τ−−1

n=0 I(Sn ∈ A) and U−(A) = E∑τ+−1

n=0 I(Sn ∈ A)(c) Eτ− = ||U+|| = (1− ||G+||)−1 and Eτ+ = ||U−|| = (1− |||G−|)−1

Proof.

(a) We use Lemma 1.2. For every j = 1, . . . , n we have Sn > Sn−j , whichis equivalent to S∗j > 0. Since {Sn}n0 and {S∗n }n0 have the samedistribution, the statement (a) follows.

(b) If we sum up {Sn > Sk , k = 0, . . . , n − 1,Sn ∈ A} over n, we obtain{M ∈ A}. On the other hand, {Sk > 0, k = 1, . . . , n} = {τ− > n}.From (a)U+(A) = P(M ∈ A) = P(τ− > n,Sn ∈ A) = E

∑τ−−1n=0 I(Sn ∈ A).

(c) follows from (b) by letting A = [0,∞).

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1 Basic definitions

2 Ladder processes

3 Classification

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Classification of random walk

Theorem 2.4.For any random walk with F non degenerate at 0, one of the followingpossibilities occur:

Oscillating case: G+ and G− are both proper, ||G+|| = ||G−|| = 1and limSn =∞, lim Sn = −∞ a.s. Furthermore Eτ+ = Eτ− =∞Drift to +∞: G+ is proper and G− is defective, and Sn →∞ a.s.Furthermore Eτ+ = (1− ||G−||)−1



Proof.By Hewitt-Savage 0-1 law, limSn = a for some constant a ∈ [−∞,∞]. If|a| 0 and limSn = lim Sτ+(n), thus limSn =∞ if and only ifSτ+(n) is nonterminating, i.e. if and only if ||G+|| = 1.Similarly lim Sn = −∞ if and only if ||G s−|| = 1, i.e. by Proposition 1.1. ifand only if ||G−|| = 1.The statement about Eτ+ follows directly from the Theorem 2.3.By LLN Sn/n

a.s.→ EX . If EX > 0, then eventually Sn > 0 and thusSn →∞ a.s.

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Occupation measure

Define

U =∞∑

n=0F ∗n.

So for any Borel set A ⊆ R

U(A) =∞∑

n=0P(Sn ∈ A) = E

∞∑n=0

I(Sn ∈ A)

is the expected number of visits of the random walk to A. We call thismeasure occupation measure.

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Lemma 2.5.If F is nonlattice, then supp(U) = R. If F is aperiodic on Z thensupp(U) = Z.

If F is lattice, say aperiodic on Z, then Sn may be viewed as Markovchain on Z and irreducibility follows from Lemma 2.5. We haverecurrence if

∑∞n=0 P(Sn = 0) =∞ and transience otherwise.

In nonlattice case similar result holds

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Classification via occupation measure

Theorem 2.6.For any nonlattice random walk, one of the following possibilities occur:

Transient case: For any bounded interval J , we have U(J)



Appendix

Theorem V.2.9.In the zero-delayed case with ||F || < 1, the distribution of M is(1− ||F ||)U.

Theorem V.4.6.Let {Bt}t≥0 be the forward recurrence time process in a (possiblydelayed) renewal process with interarrival distribution F . ThenP(Bt ≤ ξ)→ F0(ξ) for all ξ > 0. In particular, if µ < 1 then Bt

d→ F0.

Hewitt-Savage 0-1 lawLet {Xn} be a sequence of i.i.d. random variables. Then the sigma fieldof exchangeable events � is trivial.Use: Let Sn =

∑ni=1 Xn be a random walk. Then the probability that Sn

visits one state infinitely often is zero or one.

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References

Asmunssen, SorenApplied probability and queuesSpringer, 2003

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Thank you for your attention.

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Basic definitionsLadder processesClassification

random walks applied probability and queues by s....

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