On the distribution of crossings in a generalized random walk

This paper deals with the derivation of the probability distribution of the rank order statistic N¹_μ,n(r), the number of crossings of heightr(≥0) in a generalized random walk with steps 1 and ?μ by using the modified Dwass technique.     

Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

A semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζ_n) → ∞ when the random variable ζ_n has a generalized beta distribution with parameters (s, S, α, β); ?α, β > 1,?0? ? s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.