The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands (Q1706462)

The local asymptotic estimation for the supremum of a random walk is considered for the distributions of the summands of the random walk which are long-tailed and generalized subexponential distributions. It is shown that for any \(0<T<\infty\) it is true that \(\lim \inf W(x+\Delta_T)(\overline{F}(x))^{-1}=\mu^{-1}T\) for the random walk with long-tailed distribution \(F\). Beside this, it is shown that this condition is satisfied for the long-tailed and generalized strong subexponential distributions. At the end of the paper some applications of the presented results are given.