**Sankhya:
The Indian Journal of Statistics**

2006, Volume 68, Pt. 4, 221--236

**Nonuniform Rates of Convergence to Normality**

Ratan Dasgupta, Indian Statistical Institute, Kolkata

SUMMARY. Nonuniform rates of convergence to normality are studied for standardized sample sum of independent random variables in a triangular array when $m$th moment of the variables is of order $L^m \exp(\gamma m \log m), \; L > 0, 0 < \gamma < 1, \; \forall m > 1$; equivalently, $\sup_{n \geq 1} n^{-1} \sum^n_{i=1} E \exp (s | X_{ni} | ^{1 / \gamma}) < \infty$, for some $s > 0$. This assumption goes beyond the existence of moment generating functions of individual random variables. As $0 < \gamma < 1$, one gets a clear picture of the role of $\gamma$ on rates of convergence, while one moves from the assumption of existence of the moment generating functions of the random variables to the boundedness of the random variables, by varying $\gamma$. Linnik (1961) considered convergence rates in iid setup with variables having moment generating functions at the most. The general results considered in the present paper reduce to those of Dasgupta (1992) in the special case $\gamma = 1/2$. The nonuniform bounds are used to obtain rates of moment type convergences and $L_p$ version of Berry-Esseen theorem. An upper bound for the tail probability of standardized sample sum being greater than $t$ is computed. For $0 < \gamma < 1/2$ and $t$ large, this probability is shown to have a faster rate of decrease than normal tail probability. The results are extended to general nonlinear statistics and linear process.

*AMS (2000) subject classification. *Primary 60F99; secondary 60F05, 60F10, 60G50.

*Key words and phrases. *Nonuniform rates, $L_p$ version of Berry-Esseen theorem, tail probability, linear process.