General theorems on “Little-o” rates of closeness of two weighted sums of independent Hilbert space valued random variables with applications

General Limit Theorems with $o$-Rates and Markov Processes under Pseudo-Moment Conditions

“…As faras the authors are aware, there are no comparable general limit ,theorems with rates in the literature. • The type of convergence considered in this paper is more general than that in BUTZER, HAHN and ROECRERATH [10], or in BUTZER and SCHULZ [13] where the dependency structure is of type of martingale difference sequences. In fact; the convergence considered in (1.1) is that for the difference f /(x + u) dFT,,(x) -f /(x ± u) dFz(x)…”

“…The latter. were generalized to the case of not necessarily independent random variables which form martingale difference sequences or al-rays in ' BuJTZER, HAHN and ROECKERATH [10,11], and to more general types of dependent random variables in BUTZER and ScHuLz [13][14][15], 'as well as to arbitrary sequences for the particular case of identically distributed random variables in BUTZER and KmscIEFINK [12]. As faras the authors are aware, there are no comparable general limit ,theorems with rates in the literature.…”