Null recurrence and transience of random difference equations in the contractive case

Abstract

Given a sequence (M k , Q k ) k≥1 of independent, identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (X n ) n≥0 , defined by the random difference equation X n = M n X n−1 + Q n for n ≥ 1, where X 0 is independent of (M k , Q k ) k≥1 . Criteria for the null recurrence/transience are provided in the situation where (X n ) n≥0 is contractive in the sense that M 1 · . . . · M n → 0 a.s., yet occasional large values of the Q n overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (X n ) n≥0 under the sole assumption that this chain is locally contractive and recurrent.

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

“…Also, for each n ∈ N and integer 1 ≤ k ≤ n, set I k,n := ((k − 1)/n, k/n] and let F k,n denote the σ-algebra generated by (H s− , Z (1) s ) 0≤s≤k/n (we also denote by F 0,n the trivial σ-algebra). Recalling that Z (1) is a drift-free subordinator we write…”

“…A major part of the recent book [23] is concerned with the so defined perturbed random walks, both multiplicative and additive. We refer to the cited book for numerous applications of these random sequences and to [1,15,17,24,25] for more recent contributions.…”

Rate of escape of conditioned Brownian motion

“…where the sequence (α n , β n ) n is i.i.d. with positive coefficients, α n < 1 and β n with logarithmic tails, P(β 1 > t) ∼ c/ ln t for large t. Although autoregressive processes AR(1) of the type (1.4) are usually addressed with exponential or power-law tail for β n , see [5], the case of logarithmic tail has been also considered, see [15], [31], [3], and also both papers [1] and [32] for a recent account. Interestingly, our model is critical in the perspective of the Markov chain S n , in the sense that the actual value of the constant c is precisely the transition from recurrence to transience for the chain.…”

“…and satisfy a < 0 (contractive case), 0 < b < ∞ (very heavy tail). Following [1] and [32], this prevents the Markov chain S n to be positive recurrent: though the contraction brings stability to the process, yet occasional large values of β n overcompensate this behavior so that positive recurrence fails to hold. In our case, we easily check from (3.…”

“…is "big enough and well distributed in space", then the proportion of visited sites is approximately uniformly distributed on [0,1]. In [20] the explicit formula for the Green function is obtained, and a survey is given in Chapter 4 of [21].…”