Convergence of Random Processes and Limit Theorems in Probability Theory

The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).