Limit theorem for the Moment at Which a Random Walk Attains Its Maximum at a Fixed Level in the Region of Tempered Deviations

We consider a random walk with zero mean and finite variance whose steps are arithmetic. The limit arcsine law for the time at which a walk attains its maximum is well known. In this paper, we consider the distribution of the moment of attaining the maximum under the assumption that the maximum value itself is fixed. We show that, in the case of a tempered 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 were obtained in the nonlattice case.