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A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering

Authors: Mark Finkelstein and Howard G. Tucker
Journal: Proc. Amer. Math. Soc. 107 (1989), 1061-1070
MSC: Primary 60F05; Secondary 60G50
MathSciNet review: 993749
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Abstract: Let $ \left\{ {{X_n}} \right\}$ be a sequence of independent, identically distributed (i.i.d.) random variables with common mean $ \mu \ne 0$ and variance $ {\sigma ^2} > 0$. Let $ \left\{ {{S_n}} \right\}$ be a sequence of nonnegative integer-valued random variables such that for each $ n$ the random variables $ {S_n},{X_1},{X_2}, \ldots $ are independent. Then $ ({X_1} + \cdots + {X_{{S_n}}} - n\mu )/\sqrt {n{\sigma ^2}} \mathop \to \limits^L $ if and only if $ Z$, in which case the distribution of $ ({S_n} - n)/\sqrt n \mathop \to \limits^L $ is that of $ U$, where $ Z$ and $ X + Y$ are independent random variables, $ X$ being $ Y$ and $ X$ having the same distribution as $ N\left( {0,1} \right)$.

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Keywords: Random sums of random variables, nonrandom centering, convergence in law
Article copyright: © Copyright 1989 American Mathematical Society

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