Convergence of weighted partial sums when the limiting distribution is not necessarily Radon (Q1593627)

Let \((X_i)\) be a sequence of i.i.d. random variables with \(EX_i=0\) and Var\(X_i=1\). The partial sum process associated with \((X_i)\) is defined as \(S_n(t)=\sum _{i=1}^{[nt]}X_i\), \(n\geq 1.\) Further, let \(T\subset (0,\infty)\) be an interval and let \(w\) be any positive locally finite function on \(T\) (weight function). Then \[ {\mathbb B}_w= \Bigl\{f\in {\mathbb R}^T\mid \|f\|_w=\sup _T |f(t)|w(t)<\infty \Bigr\} \] equipped with the \(\|.\|_w\) norm is a Banach space. Let \(\mathbb {D}_w\) denote the closed subspace of \(\mathbb {B}_w\) consisting of all right continuous functions having left-hand-side limits. The main result of the paper is the following: Let \(w\) be a weight function such that the \(\sigma \)-algebra generated by balls in \(\mathbb {D}_w\) is contained in \(\text{Borel}^T\cap \mathbb {D}_w\). Then the following are equivalent: (i) \(\|B\|_w<\infty \) a.s. for any Brownian motion \(B\) on \(\mathbb {B}_w\) (the corresponding Wiener measure need not be Radon); (ii) any partial sum process induces a sequence of \(\mathbb {B}_w\)-valued random variables \(S_n\) such that \(\lim _{n\to \infty} \frac {\|S_n\|_w}{\sqrt {n}}=\|B\|_w\) in distribution; (iii) any partial sum process induces a sequence of \(\mathbb {B}_w\)-valued random variables \(S_n\) such that \(\{\|S_n\|_w/\sqrt {n}\}\) is bounded in probability.