Multiple range of random walk up to the time of exit from an interval

Consider the p-multiple range R (p)N which counts the number of points visited exactly p ≥ 1 times by a one-dimensional simple symmetric random walk starting at [αN ], for α ∈ (0, 1), up to the time of exit from D N = {0, 1, . . ., N }.We show that R (p) N / log(N ) converges weakly to the law of an exponential random variable with mean 1/2.Moreover, we show, by the method of moments, that the collection of scaled multiple ranges {R (p) N / log(N ) : p ≥ 1} in the limit is totally correlated.