THE INFLUENCE ON THE ASYMPTOTICS OF THE RANDOM WALKS CAUSED BY THE VARIATION OF THE INCREMENTS

Abstract

Let {X_n : n ≥ 1} be a sequence of independent and identically distributed random variables with a common mean µ ∈ (−∞, 0) and {S_n : n ≥ 0} be the random walk generated by {X_n : n ≥ 1}.  For any ε ∈ (0,−µ), let {\(S_n ^{(±ε)}\) : n ≥ 0} be the random walks generated by {X_n ± ε : n ≥ 1}.  This paper considers the limits of P \(\left( \underset{n≥0}{sup} \, S_n^{(±ε)} > x \right)\) / P \(\left( \underset{n≥0}{sup} \, S_n > x \right)\) as x \(\to\) \(\infty\).