On the reflected random walk on R+

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ = ∫Rydρ(y) < 0. Consider the random walk on R+ starting at x ∈ R+ and defined for any n ∈N by \begin{eqnarray*} \left\{\begin{array}{rl} X_0&=x \\ X_{n+1} & = |X_n+Y_{n+1}| \end{array}\right. \end{eqnarray*} where (Yn) is an iid sequence of law ρ. We denote P the Markov operator associated to this random walk and, for any borelian bounded function f on R+, we call Poisson’s equation the equation f = g − Pg with unknown function g. In this paper, we prove that under a regularity condition on ρ and f, there is a solution to Poisson’s equation converging to 0 at infinity. Then, we use this result to prove the functional central limit theorem and it’s almost-sure version.

On the local limit theorem for non-uniformly ergodic Markov chains

Using an approach similar to that of Guivarc'h and Hardy (1988), we show that the local limit theorem holds for a Markov chain on a countable state space, with non-uniform ergodicity, when the recurrence is fast enough. We present a detailed study of a typical example, the reflected random walk on the positive half-line with negative mean and finite exponential moment. The results can be extended to some random walks on ℕ.

On the local limit theorem for non-uniformly ergodic Markov chains

Using an approach similar to that of Guivarc'h and Hardy (1988), we show that the local limit theorem holds for a Markov chain on a countable state space, with non-uniform ergodicity, when the recurrence is fast enough. We present a detailed study of a typical example, the reflected random walk on the positive half-line with negative mean and finite exponential moment. The results can be extended to some random walks on ℕ.

A coupling proof of weak convergence

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.

A coupling proof of weak convergence

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.