Some Generalized Central Limit Theorems for Weighted Sums with Infinite Variance

Abstract

For i.i.d. random variables {Y, Y_n, n⩾ 1} with EY²=∞ and nonzero constants {a_n, n⩾1}, sufficient and, separately, necessary conditions are given for {a_n Y_n, n⩾1} to obey a generalized central limit theorem $\left({\displaystyle \sum _{j=1}^n\hspace{0.17em}{a}_j\hspace{0.17em}{Y}_j\hspace{0.17em}-\hspace{0.17em}An}\right)\cancel{}{B}_n\stackrel{d}{\to }\hspace{0.17em}N\left(0,1\right)$ for suitable constants {A_n, n⩾1} and {B_n > 0, n⩾1}. The norming constants {B_n, n⩾1} are defined using the sequence $\left\{{\displaystyle \sum _{j=1}^n\hspace{0.17em}{a}_j^2,\hspace{0.17em}n\hspace{0.17em}⩾1}\right\}$ and the distribution of Y. Moreover, it is shown that if p{|Y|> y} is regularly varying with exponent-2, then the centering constants may be taken to be $A_n=\hspace{0.17em}{\displaystyle \sum _{j=1}^n\hspace{0.17em}{a}_j}\hspace{0.17em}EY,\hspace{0.17em}n⩾1$ . A famous result of Feller (1935), Khintchine (1935), and Lévy (1935) is obtained in the special case a_n ≡ 1.