The random variables (r.vs.) X , . i E S : = ( 1 , 2 ...I, to he dealt uith are measurable mappings from a probability space (9. 91, P ) into a measure space (B, %), R being a RANACH space with a countable hasis (e,),,,, and ' 3 the o-algebra of 13orel sets of B. The type of convergence to be mainly considered is the convergence in distribution of the B-valued partial sums Sn=c Xi, thus the weak convergence o f the distrihutions PAa of S, to that of a limiting r.v.Z, denoted hy Pz. This means that n i -1
for each f <C', ( =set. of bounded continuous real valued functions on B; c.f. (2.2)).
Actually randomly indexed sums are to be considered. If (Nn)ncx is a sequence of r.m. defined on (9, %. P) with values in S s u c h that Nn-m in proha1)ility for n-w. then the random sum Savn, defined by converges in distribution to Z if. for each fECs, lyow a large number of papers is concerned with tlo-called transmission theorems for r.vs. with values in a separable metric space X. Assuming that (1.1) is valid for B = S , t,hey are concerned with conditions which must be posed upon the r.vs. Xi, ic S. and N,, n 6 S . such that (1.3) holds. See for infitance 1281. [21)]. 1253, [31], LlS], [l], [12]. [35].
The major aim of t,his paper is to deduce not just convergence assertions of type (1.3) directly. i.e.. without a ~u m i n g the validity of (1.1) or corresponding conditions, but bo examine the rate of convergence for assertions of type (1.3). In this respect transmission theorems with built-in rates for metric-space-valued r.vs. are not known.