CONDITIONAL LIMIT THEOREMS FOR THE TERMS OF A RANDOM WALK REVISITED

In this paper we derive limit theorems for the conditional distribution of X-1 given S-n = s(n) as n -> infinity, 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-X1 vertical bar Sn/n=s in, to a distribution associated to that of X-1 and of P-nX1 vertical bar 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.