egin{abstract} hspace*{1cm} Let $X_{1}$@, $X_{2}$@,$cdots$@, $X_{n}$ be a sequence of independent indentically distributed random variables ( i@. i@. d@.) and $S_{n}=X_{1}+X_{2}+...+X_{n}$@. Denote $lambda(varepsilon)=displaystylesum_{n=1}^{infty}P(|S_{n}|geq nvarepsilon)$@. O.I. Klesov proved that if $EX_{1}=0$, $EX_{1}^{2}=sigma ^{2} eq 0$, $E|X_{1}|^{3}<infty$, then $displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$. In this thesis, it is shown that if $EX_{1}=0$, $EX_{1}^{2}=sigma ^{2} eq 0$, $E|X_{1}|^{2+delta}<infty$ for some $displaystyledeltain(frac{1}{2},1]$, then $displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$. end{abstract}

egin{abstract} hspace{1cm}Let $X_{1}$, $X_{2}$, $cdots$, $X_{n}$ be a sequence of independent indentically distributed random variables ( i. i. d.) and $displaystyle S_{n}=X_{1}+X_{2} +cdots X_{n}$. Denote $displaystylelambda(varepsilon)=sum^{infty}_{n=1}P{left|S_{n} ight|geq nvarepsilon}$, the convergence rate of $displaystylelambda(varepsilon)$ is studied. O.I. Klesov proved that if $E|X_{1}|^{3}$ exists, then $displaystyle varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}}) ightarrow 0$. In this thesis, we show that if $E|X_{1}|^{2+delta}<infty$ for some $displaystyle deltain(frac{sqrt{7}-1}{3},1]$, the result of O.I. Klesov still holds. end{abstract}