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Abstract

For nonzero constants {a_n, n ⩾1} and i.i.d. random variables {Y, Y_n n⩾1} with EY² = ∞ and $_{y \to \infty}^{\lim} y^{2} P {| Y | > y} / G (y) = 0 w h e r e G (y) = \int_{o}^{y} 2 t P {| Y | > t} d t, y ⩾ 0$ Adler and Rosalsky (1989) proved a generalzed central limit theorem $(\sum_{j = 1}^{m} a j Y j - A n) / B_{n} \overset{d}{\to} N (0, 1) w h e r e {A n, n ⩾ 1} a n d {B n, n ⩾ 1}$ are suitable numerical sequences. In addition, Adler and Rosalsky(l989) showed that the centering constants {A_n, n⩾1} can be taken to be $A_{n} = \sum_{j = 1}^{n} a_{j} E y, n ⩾ 1$ if the condition $_{y \to \infty}^{\lim} y 2 P {| Y | > y} / G (y) = 0$ is replaced by the stronger condition that p{ | Y | > y} is regularly varying with exponent - 2. In this note, it is shown that the generalized central limit theorem does indeed hold with $A_{n} = \sum_{j = 1}^{n} a_{j} E Y, n ⩾ 1$ without any strengthening of the condition $_{y \to \infty}^{\lim} y^{2} P {| Y | > y} / G (y) = 0$ .

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A Note on the Centering Constants in the Adler-Rosalsky Central Limit Theorem

Abstract

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