Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

In this paper we derive limit theorems for the conditional distribution of X1 given Sn = Sn as n ? 8, where the Xi are independent and identically distributed (i.i.d.) random variables, Sn = X1 + ··· + Xn, and sn/n converges or sn = s is constant. We obtain convergence in total variation of P... to a distribution associated to that of X1 and of P... to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

E. Schulte-Geers

Papers

Some results on the joint distribution of the renewal Epochs prior to a given time instant