On convergence of normalized partial sums

Abstract

Let ($X_n$) be a sequence of random variables and $S_n = \sum^n_{i=1}X_i$. Pyke and Root (1968) proved that if ($X_n$) is a sequence of i.i.d. random variables with $E│X_1│^r＜∞$, then ($S_n-ES_n)/n^{1/r} → 0 a.s. and in $L^r$, 1≤r＜2. Chatterji(1969) extended this result to the case where the $X_n``s$ are dominated in distribution by a random variable X with $E│X│^r＜∞$. Recently Etemadi(1981, 1983) proved that Pyke and Root``s result under the weaker condition of pairwise independence when r=1. In this thesis, we prove that if ($X_n$) are uncorrelated and their second moments have a common bound, then ($S_n-ES_n)/n^{1/r} → 0 a.s. for 1≤r＜2. We also prove that if ($X_n$) are pairwise independent and dominated in distribution by a random variable X with $E(│X│^r(log^+│X│^r)^2)＜∞$, then ($S_n-ES_n)/n^{1/r} → 0 a.s. for $1<r<2$. Finally, we prove that if ($X_n$) are pairwise independent and dominated in distribution by a random variable X with $E│X│^r＜∞, then $(S_n-ES_n)/n^{1/r}$ → 0 in $L^1$ for 1≤r<2$.