Limit Theorem for the Moment of Maximum of a Random Walk Reaching a Fixed Level in the Region of Moderate Deviations

We consider a random walk with zero mean and finite variance whose steps are arithmetic. The arcsine law for the time the walk reaches its maximum is well known. In this paper, we consider the distribution of the moment of reaching the maximum under the assumption that the maximum value itself is fixed. We show that, in the case of a moderate deviation of the maximum, the distribution of the moment of the maximum with appropriate normalization converges to the chi-square distribution with one degree of freedom. Similar results are obtained in the nonlattice case.