The Length of the Longest Head-Run in a Model with Long Range Dependence

Abstract

In this paper, we construct stationary sequences of random variables {χi: i≥0} taking values ±1 with probability 1/2 and we prove an Erdos–Renyi law of large numbers for the length of the longest run of consecutive +1's in the sample {χ0,..., χn}. Our model, which is called random walk in random scenery, exhibits long-range, positive dependence.