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A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering. (English) Zbl 0682.60017

Summary: Let \(\{X_ n\}\) be a sequence of independent, identically distributed (i.i.d.) random variables with common mean \(\mu\neq 0\) and variance \(\sigma^ 2>0\). Let \(\{S_ n\}\) be a sequence of nonnegative integer- valued random variables such that for each n the random variables \(S_ n\), \(X_ 1\), \(X_ 2,..\). are independent. Then \[ (X_ 1+...+X_{S_ n}-n\mu)/\sqrt{n\sigma^ 2}\to^{{\mathcal L}}(some)\quad Z\quad if\quad and\quad only\quad if\quad (S_ n-n)/\sqrt{n}\to^{{\mathcal L}}\quad (some)\quad U, \] in which case the distribution of Z is that of \(X+Y\), where X and Y are independent random variables, X being N(0,1) and Y having the same distribution as \(\mu\) U/\(\sigma\).

MSC:

60F05 Central limit and other weak theorems
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