PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES
PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES
Let {<TEX>$X,\;X_n;n{\geq}1$</TEX>} be a sequence of i.i.d. random variables. Set <TEX>$S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1$</TEX>. Then we obtain that for any -1<b<1/2, <TEX>$\lim\limits_{{\varepsilon}{\searrow}0}\;{\varepsilon}^{2b+2}\sum\limits_{n=1}^\infty\;{\frac {(log\;n)^b}{n^{3/2}}\;E\{M_n-{\varepsilon}{\sigma}\sqrt{n\;log\;n\}+=\frac{2\sigma}{(b+1)(2b+3)}\;E|N|^{2b+3}\sum\limits_{k=0}^\infty\;{\frac{(-1)^k}{(2k+1)^{2b+3}$</TEX> if and only if EX=0 and <TEX>$EX^2={\sigma}^2</TEX><TEX><</TEX><TEX>{\infty}$</TEX>.