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

DOI:
https://doi.org/10.1090/S0002-9939-1989-0993749-9

MathSciNet review:
993749

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of independent, identically distributed (i.i.d.) random variables with common mean and variance . Let be a sequence of nonnegative integer-valued random variables such that for each the random variables are independent. Then if and only if , in which case the distribution of is that of , where and are independent random variables, being and having the same distribution as .

**[1]**Harald Cramér,*Mathematical Methods of Statistics*, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946. MR**0016588****[2]**B. V. Gnedenko and H. Fahim,*On a transfer theorem*, Soviet Math. Dokl.**10**(1969), 769-772.**[3]**Zdzisław Rychlik,*A central limit theorem for sums of a random number of independent random variables*, Colloq. Math.**35**(1976), no. 1, 147–158. MR**0410866****[4]**A. Rényi,*On the central limit theorem for the sum of a random number of independent random variables*, Acta Math. Acad. Sci. Hungar.**11**(1960), 97–102 (unbound insert) (English, with Russian summary). MR**0115204**, https://doi.org/10.1007/BF02020627**[5]**J. G. Shanthikumar and U. Sumita,*A central limit theorem for random sums of random variables*, Oper. Res. Lett.**3**(1984), no. 3, 153–155. MR**761508**, https://doi.org/10.1016/0167-6377(84)90008-7**[6]**Howard G. Tucker,*A graduate course in probability*, Probability and Mathematical Statistics, Vol. 2, Academic Press, Inc., New York-London, 1967. MR**0221541**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0993749-9

Keywords:
Random sums of random variables,
nonrandom centering,
convergence in law

Article copyright:
© Copyright 1989
American Mathematical Society