Patrick Billingsley

Weak convergence of measures : applications in probability

Weak convergence Random elements and convergence in distribution Prokhorovs theorem The Space C A maximal Inequality Tightness Limit theorems.

Convergence of types in k-space

SummaryIf random variables Xn converge in distribution to a nondegenerate random variable X, and if transformations anXn+bn of them, with an>0, also converge in distribution to X, then an→1 and bn→0. Moreover, the group of transformations x→ax+b that preserve the distribution of a nondegenerate random variable consists either of the identity alone, or of the identity together with the reflection through some point. For proofs see [4], [5], or [6]. This paper gives the corresponding results for k-dimensional random vectors.

Prime Numbers and Brownian Motion

(1973). Prime Numbers and Brownian Motion. The American Mathematical Monthly: Vol. 80, No. 10, pp. 1099-1115.

Convergence of Probability Measures.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.