Records from stationary observations subject to a random trend

Abstract

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Y_n = X_n + T_n, n ≥ 1, where (X_n)_{n ∈ Z} is a stationary ergodic sequence of random variables and (T_n)_{n ≥ 1} is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.