Limit theorems for affine Markov walks conditioned to stay positive

Abstract

Consider the real Markov walk S n = X 1 + · · · + X n with increments (X n ) n 1 defined by a stochastic recursion starting at X 0 = x. For a starting point y > 0 denote by τ y the exit time of the process (y + S n ) n 1 from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event τ y n and of the conditional law of y + S n given τ y n as n → +∞.

Conditioned local limit theorems for random walks defined on finite Markov chains

“…Important progress has been achieved by employing a new approach based on the existence of the harmonic function in Varopoulos [27], [28], Eichelbacher and König [10] and recently by Denisov and Wachtel [6,7,8]. In this line Grama, Le Page and Peigné [16] and the authors in [12], [13] have studied sums of functions defined on Markov chains under spectral gap assumptions. The goal of the present paper is to complete these investigations by establishing local limit theorems for random walks defined on finite Markov chains and conditioned to stay positive.…”

First-passage times for random walks with nonidentically distributed increments

“…• Limit theorems for Markov walks conditioned to stay positive, see [11] and [12]. Besides asymptotic results we can use the universality approach to construct conditioned processes and prove functional limit theorems for conditioned process.…”

“…However, the main restriction of the method was the necessity to use a strong coupling, which is difficult to prove and is rarely available. For example, papers [13], [11] and [12] depend on [14], where an FCLT with a rate of convergence (strong coupling) was proved. The present paper deals with this deficiency and allows one to use directly the FCLT instead of the strong coupling.…”

“…Theorems 1 and 2 state that for any random walk belonging to the domain of attraction of the Brownian motion and for any boundary sequence g n = o(B n ), with necessary condition (6), we have universal limiting behaviour of conditional distributions. In (11) we also have the universal leading term: B −1 n . And dependence on the boundary {g n } and on the distribution of the increments {X k } concentrates in the function U g only.…”

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

“…This allowed a more adequate modeling and turned out to be very fruitful from the practical as well as from the mathematical points of view. The recent advances in the study of conditioned limit theorems for sums of functions defined on Markov chains in [14], [11], [13] and [12] open the way to treat some unsolved questions in the case of Markovian environments. The problem we are interested here is to study the asymptotic behaviour of the survival probability.…”