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

In this paper, we construct stationary sequences of random variables {X i : 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 +1ʹs in the sample {X0,..., Xn }. Our model, which is called random walk in random scenery, exhibits long-range, positive dependence.