Conditional Limit Theorems for the Terms of a Random Walk Revisited

Abstract

In this paper we derive limit theorems for the conditional distribution of X 1 given S n =s n as n→ ∞, where the X i are independent and identically distributed (i.i.d.) random variables, S n =X 1+··· +X n , and s n /n converges or s n ≡ s is constant. We obtain convergence in total variation of P X 1∣ S n /n=s to a distribution associated to that of X 1 and of P nX 1∣ S n =s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.