Nonuniform Rates of Convergence to Noimality
Ratan Dasgupta · 2016
Nonuniform rates of convergence to normality are studied for standardized sample sum of independent random variables in a triangular array when rath moment of variables is of order Lm expira logra), L > 0,0 1; equivalently, supn>1 n~l J27=i Eexp(s\Xni\1/l) 0. This assumption goes beyond existence of moment generating functions of individual random variables. As 0 < 7 < 1, one gets a clear picture of role of 7 on rates of convergence, while one moves from assumption of existence of moment generating functions of the random variables to boundedness of random variables, by varying 7. Lin nik (1961) considered convergence rates in iid setup with variables having moment generating functions at most. The general results considered in present paper reduce to those of Dasgupta (1992) in special case 7 = 1/2. The nonuniform bounds are used to obtain rates of moment type convergences and Lp version of Berry-Esseen theorem. An upper bound for tail probability of standardized sample sum being greater than t is com puted. For 0 < 7 < 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.