Ladder Epochs Research Articles

Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

Abstract Let (Mn,Sn)n≥0 be a Markov random walk with positive recurrent driving chain (Mn)n≥0 on the countable state space 𝒮 with stationary distribution π. Suppose also that lim supn→∞Sn=∞ almost surely, so that the walk has almost-sure finite strictly ascending ladder epochs σn>. Recurrence properties of the ladder chain (Mσn>)n≥0 and a closely related excursion chain are studied. We give a necessary and sufficient condition for the recurrence of (Mσn>)n≥0 and further show that this chain is positive recurrent with stationary distribution π> and 𝔼π>σ1><∞ if and only if an associated Markov random walk (𝑀̂n,𝑆̂n)n≥0, obtained by time reversal and called the dual of (Mn,Sn)n≥0, is positive divergent, i.e. 𝑆̂n→∞ almost surely. Simple expressions for π> are also provided. Our arguments make use of coupling, Palm duality theory, and Wiener‒Hopf factorization for Markov random walks with discrete driving chain.

Notes on random walks in the Cauchy domain of attraction

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an $\alpha$-stable law with $\alpha \in(0,2)$, so we also present results in that case, since many of them seem to be missing in the literature.

Records from stationary observations subject to a random trend

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Y n = X n + T n , n ≥ 1, where (X n ) n ∈ Z is a stationary ergodic sequence of random variables and (T n ) n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

Records from stationary observations subject to a random trend

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n ∈ Z is a stationary ergodic sequence of random variables and (Tn)n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

A note on summability of ladder heights and the distributions of ladder epochs for random walks

This paper concerns a recurrent random walk on the real line R and obtains a purely analytic result concerning the characteristic function, which is useful for dealing with some problems of probabilistic interest for the walk of infinite variance: it reduces them to the case when the increment variable X takes only values from { … , − 2 , − 1 , 0 , 1 } . Under the finite expectation of ascending ladder height of the walk, it is shown that given a constant 1 < α < 2 and a slowly varying function L ( x ) at infinity, P [ X < − x ] ∼ − x − α / Γ ( 1 − α ) L ( x ) ( x → ∞ ) if and only if P [ T > n ] ∼ n − 1 + 1 / α / Γ ( α ) L α ∗ ( n ) , where L α ∗ is a de Bruijn α -conjugate of L and T denotes the first epoch when the walk hits ( − ∞ , 0 ] . Analogous results are obtained in the cases α = 1 or 2. The method also provides another derivation of Chow’s integrability criterion for the expectation of the ladder height to be finite.

Transition phenomena for ladder epochs of random walks with small negative drift

For a family of random walks {S (a)} satisfying E S 1 (a)=-a&amp;lt;0, we consider ladder epochs τ(a)=min {k≥1: S k (a)&amp;lt;0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)&amp;gt;n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

Transition phenomena for ladder epochs of random walks with small negative drift

For a family of random walks {S(a)} satisfying E S1(a)=-a&lt;0, we consider ladder epochs τ(a)=min {k≥1: Sk(a)&lt;0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)&gt;n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Markov renewal theory for stationary ( m+1)-block factors: convergence rate results

This article continues work by Alsmeyer and Hoefs (Markov Process Relat. Fields 7 (2001) 325–348) on random walks ( S n ) n⩾0 whose increments X n are ( m+1)-block factors of the form ϕ( Y n− m ,…, Y n ) for i.i.d. random variables Y − m , Y − m+1 ,… taking values in an arbitrary measurable space ( S, S) . Defining M n =( Y n− m ,…, Y n ) for n⩾0, which is a Harris ergodic Markov chain, the sequence ( M n , S n ) n⩾0 constitutes a Markov random walk with stationary drift μ= E F m+1 X 1 where F denotes the distribution of the Y n 's. Suppose μ>0, let ( σ n ) n⩾0 be the sequence of strictly ascending ladder epochs associated with ( M n , S n ) n⩾0 and let ( M σ n , S σ n ) n⩾0 , ( M σ n , σ n ) n⩾0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with ( M n , S n ) n⩾0 and the former two sequences are denoted U λ , U λ > and V λ >, respectively, where λ is an arbitrary initial distribution for ( M 0, S 0). Given the basic sequence ( M n , S n ) n⩾0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of U λ , U λ > and V λ > under natural moment conditions. Proofs are based on a suitable reduction to standard renewal theory by finding an appropriate imbedded regeneration scheme and coupling. Considerable work is further spent on necessary moment results.

A new proof of finite moment conditions for GI/G/1 busy periods

A generalization of the GI/G/1 queue is considered where the service time of the nth customer and the inter-arrival time between arrivalsn andn+1 may be dependent random variables. New proofs are obtained of finite moment conditions for busy periods and the ladder epochs of a corresponding random walk. The method of proof, which is much different from the usual ones, directly relates busy period moments to virtual and actual delay moments.

On the exact asymptotic behaviour of the distribution of ladder epochs

Let T + denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E( X)=0, E( X 2)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, PT +⩾n∽n 1 β−1 L +(n) as n→∞, with 1< β<2 and L + slowly varying, if and only if P{X⩾x}∽ 1/{x βL(x)} as x→+∞, with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L + and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.

Some inequalities on the distribution of ladder epochs

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.

Some inequalities on the distribution of ladder epochs

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.

Dual Pairs of Stopping Times for Random Walk

A definition of duality for pairs of stopping times of any random walk is motivated by the duality relation of ascending and descending ladder epochs $N, \bar{N}$ of random walk in $R^1$. Dual pairs share several of the properties of the pair $N, \bar{N}$.

Uniform Tauberian Theorems and their Applications to Renewal Theory and First Passage Problems

In this paper, we prove an analogue of the classical renewal theorem for the case where there is no drift. Our proof depends on a uniform version of Spitzer's well-known theorem on ladder epochs and ladder variables, and we obtain this uniform result by using uniform Tauberian theorems. Some further applications of these uniform Tauberian theorems to other problems in renewal theory and first passage times are also given.

Some Limit Theorems for Large Deviations

Some Limit Theorems for Large Deviations