Conditional limit theorems for the terms of a random walk revisited

Abstract

In this paper we derive limit theorems for the conditional distribution of X₁ given S_n=s_n as n→ ∞, where the X_i are independent and identically distributed (i.i.d.) random variables, S_n=X₁+··· +X_n, and s_n/n converges or s_n ≡ s is constant. We obtain convergence in total variation of P_{X1∣ Sn/n=s} to a distribution associated to that of X₁ and of P_{nX1∣ Sn=s} to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.