Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 3 , pp. 399-415

A NON-UNIFORM RATE OF CONVERGENCE IN THE LOCAL LIMIT THEOREM FOR INDEPENDENT RANDOM VARIABLES

By

KIRKIRE PRASHANT DATTATRAYA, University of Baroda and Operations Research Group

SUMMARY. Let $\{ X_n \} , \; n = 1,2,3,... $ be a sequence of independent random variables with corresponding sequence of distribution functions $\{ G_n \}, \; n = 1,2,3,...$ . Suppose that, for every $n, G_n \in \{ F_1 , F_2 ,..., F_m \}$. Let $F_1 , F_2 ,..., F_m$ belong to the domain of normal attraction of a stable law with index $\al , 0 < \al < 2 $. Define $T_n = X_1 + \cdots + X_n $. Under fairly mild assumptions, a non-uniform rate of convergence is obtained for the density version of central limit thoerem for normalized sums $T_n$.

AMS (1991) subject classification. Primary 60F05; secondary 60E05, 62E15.

Key words and phrases. Stable law, convergence in distribution, characteristic function, central limit theorem.

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