Last passage time for the empirical mean of some mixing processes (Q1305286)

The authors study a discrete-time stochastic process \(X_1,\dots, X_n\) in \(\mathbb{R}^d\), whose empirical mean \(\overline{X}_n= n^{-1} \sum_{1\leq i\leq n}X_i\) converges to 0 almost surely. They consider two types of stochastic processes: stationary process under mixing assumptions and ergodic Markov chain. Define \(N_\varepsilon= \sup\{n\geq 1:\|\overline{X}_n \|\geq \varepsilon\}\), where \(\|x\|\) is the Euclidean norm of \(x\). The aim of this paper is to investigate the asymptotics for \(N_\varepsilon\) as \(\varepsilon\) tends to 0. Assume that the matrix \(\Lambda= \lim_{n\to\infty} \text{Var} (\sqrt{n} \overline{X}_n)\) exists. The main result of this paper is that \[ \lim_{\varepsilon\to 0}P\bigl(\varepsilon^2 N_\varepsilon\geq y\bigr)= P\Bigl(\sup_{0\leq t\leq 1}\|\Lambda^{1/2} W(t)\|\geq \sqrt{y}\Bigr), \quad y\geq 0, \] where \(W(\cdot)\) is a standard \(d\)-dimensional Brownian motion defined on \([0,1]\). Applications are given for estimation in AR models and stopping rules for simulations in Markov chain Monte Carlo methods.