A pointwise limit theorem for counting processes of perturbed random walks with an application to repeated significance tests

摘要

Hsu and Robbins (1947) introduced the concept of complete conver-gence as a complement to the Kolmogorov strong law in that they proved that Sigma(infinity)(n=1) P(|Sn| > n epsilon) < infinity provided the mean of the summands is zero and that the variance is finite. Later, Erdos proved the necessity (1949, 1950). Heyde (1975) proved that, under the same conditions, lim(epsilon)SE arrow 0 epsilon(2) Sigma(infinity)(n=1) P(|Sn| > n epsilon) = EX2, thereby opening an area of research that has been called precise asymptotics. Both results above have been extended and generalized in various directions. Kao (1978) proved a pointwise version of Heyde's result, viz. for the counting process N(epsilon) = Sigma(infinity)(n=1) 1{|Sn| > n epsilon}, he showed that lim(epsilon)SE arrow 0 epsilon N-2 (epsilon) ->(d) EX2 integral(infinity)(0) 1 {|W(u)| > u} du, where W(.) is the standard Wiener process. In this article, we prove an analog for perturbed random walks and illustrate how they enter naturally within the theory of repeated significance tests in exponential families.