On one-sided strong laws for large increments of sums
Results for increments of sums of random variables can be applied to study asymptotic behavior of increments of renewal processes, compound sum processes and several statistics. Let \(\{X_i, i\geq 1\}\) be a sequence of i.i.d. random variables with partial sums \(S_n\). The asymptotic behavior of maximal increments of partial sums over time subintervals \(a_n, a_n\leq n\), namely \(U_n= \max_{1\leq k\leq n-a_n} (S_{k+a_n} - S_k)\) is studied. Necessary and sufficient conditions for one-sided strong law of large numbers and the law of iterated logarithm for \(U_n\) are obtained. The aim of the paper is to investigate asymptotics of large increments when trajectories of \(S_n\) have large jumps only in one direction. Then the sums cannot be well approximated by a Wiener process, so strong invariance principles do not work. Large deviation techniques are used.
scientific article; zbMATH DE number 1124491
scientific article