First-passage times for random walks with nonidentically distributed increments

@article{Denisov2018FirstpassageTF,
  title={First-passage times for random walks with nonidentically distributed increments},
  author={Denis Denisov and Alexander I. Sakhanenko and Vitali Wachtel},
  journal={The Annals of Probability},
  year={2018}
}
We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander. 

First-passage time asymptotics over moving boundaries for random walk bridges

The moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Ordered exponential random walks

We study a d -dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function

Invariance principles for random walks in cones

Martin boundary of random walks in convex cones

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete

Conditioned local limit theorems for random walks on the real line

Abstract. Consider a random walk Sn = ∑n i=1 Xi with independent and identically distributed real-valued increments Xi of zero mean and finite variance. Assume that Xi is non-lattice and has a moment

First-Passage Times over Moving Boundaries for Asymptotically Stable Walks

Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves

Statistical properties of sites visited by independent random walks

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on hyper-cubic lattices and study ordering

Persistence probabilities of weighted sums of stationary Gaussian sequences

With $\{\xi_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non negative correlation function $\rho(i):=\mathbb{E}[ \xi_0\xi_i]$ and $\{\sigma(i)\}_{i\ge 1}$ a sequence of positive

First-Passage Times for Random Walks in the Triangular Array Setting

In this paper we continue our study of exit times for random walks with independent but not necessarily identical distributed increments. Our paper "First-passage times for random walks with

References

SHOWING 1-10 OF 26 REFERENCES

Exit times for integrated random walks

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. Assuming that

Conditional limit theorems for ordered random walks

In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a

Random walks in cones

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The  construction was done  under

Random walks conditioned to stay in Weyl chambers of type C and D

We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob $h$-transform.

Survival probabilities of weighted random walks

We study the asymptotic behaviour of the probability that a weighted sum of centered i.i.d. random variables Xk does not exceed a constant barrier. For regular random walks, the results follow easily

On a Functional Central Limit Theorem for Random Walks Conditioned to Stay Positive

: Let { X k : k ≧ 1 } be a sequence of i.i.d.rv with E ( X i ) = 0 and E ( X 2 i ) = σ 2 , 0 < σ 2 < ∞ . Set S n = X 1 + · · · + X n . Let Y n ( t ) be S k /σn 1 2 for t = k/n and suitably

One-Sided Boundary Crossing for Processes with Independent Increments

On etudie le comportement de la probabilite P(τ>t), t→∞, ou τ est le premier instant ou un processus X t franchit une frontiere f: τ=inf{t:X t >f(t)}

Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption

Consider a Markov chain (X n) n0 with values in the state space X. Let f be a real function on X and set S 0 = 0, S n = f (X 1) + · · · + f (X n), n 1. Let P x be the probability measure generated by

Spitzer's condition and ladder variables in random walks

SummarySpitzer's condition holds for a random walk if the probabilities ρn=P{n > 0} converge in Cèsaro mean to ϱ, where 0<ϱ<1. We answer a question which was posed both by Spitzer [12] and by Emery