A note on the extended elementary renewal theorem

Abstract

Let X ₁ ^(µ), X ₂ ^(µ), ... be an infinite sequence of independent and identically distributed random variables defined on the whole real axis and with EX₁ ^(µ) = µ > 0. Put M_n ^(µ) = max (S₀ ^(µ), S₁ ^(µ), ..., S_n ^(µ) , where S_n ^(µ) = X₁ ^(µ) + ... + X_n ^(µ) for n = 1 , 2, ... and S₀ ^(µ) = 0, and define