Abstract:
Consider a random walk with zero mean, finite variance, and arithmetic steps.
The time at which the process attains its maximum is known to obey the limit
arcsine law. In the present paper, we study the distribution of the time of
attaining the maximum under the condition that the maximum value is fixed. We
show that, with an appropriate normalization, the distribution of the time
when the process attains its maximum converges to the inverse gamma
distribution whenever the random walk attains a rare small value of the
maximum. Similar results are also obtained in the nonlattice case. The
present study supplements the earlier work by the author on a similar problem
for rare high value of the maximum.
Keywords:random walk, local limit theorem, integro-local limit theorem.
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