On Central Limit Theorems for Vector Random Measures and Measure-Valued Processes

Let $B$ be a separable Banach space. Suppose that ($F,F_i,\,i\ge 1$) is a sequence of independent identically distributed (i.i.d.) and symmetrical independently scattered (s.i.s.) $B$-valued random measures. We first establish the central limit theorem for $Y_n=\frac 1{\sqrt n} \sum_{i=1}^nF_i$ by taking the viewpoint of random linear functionals on Schwartz distribution spaces. Then, let ($X,X_i,\,i\ge 1$) be a sequence of i.i.d. symmetric $B$-valued random vectors and ($B,B_i,\,i\ge 1$) a sequence of independent standard Brownian motions on [0,1] independent of ($X,X_i,\,i\ge 1$). The central limit theorem for measure-valued processes $Z_n(t)=\frac 1{\sqrt n} \sum_{i=1}^nX_i\delta_{B_i(t)}$, $t\in [0,1]$, will be investigated in the same frame. Our main results concerning $Y_n$ differ from D. H. Thang's [Probab. Theory Related Fields, 88 (1991), pp. 1–16] in that we take into account $F$ as a whole; while the results related to $Z_n$ are extensions of I. Mitoma [Ann. Probab., 11 (1983), pp. 989–999] to random weighted mass.