Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations.~I

Yu. V. Linnik

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 2, pp. 145-163

Résumé

The independent identically distributed variables $x_1,x_2,\dots,x_n$ are supposed to have $E({x_j})=0$; $D({x_j})=\sigma^2\infty$. Denote $$Z_n=\frac{x_1+\cdots+x_n}{\sigma\sqrt n}.$$ Let $\Psi(n)\to\infty$ be some monotone function. The sequence of segments $[0,\Psi (n)]$ is called the zone of normal attraction (z. n. a.) if $$\frac{{\mathbf P(Z_n>x)}}{\frac1{\sqrt{2\pi}}\int_x^\infty{e^{-n^2/2}\,dn}}\to1$$ for $x\in[0,\Psi(n)]$; the zones $[-\Psi(n),0]$ are defined similarly as z. n. a. The zones $[0,n^\alpha];[-n^\alpha,0](\alpha>0$ constant) are called simplest. The zones such that $\Psi(n)=o(n^{1/6})$ are called “narrow”. For the random variables of the class $(d)$ (possessing a bounded continuous density) the zones $[0,\Psi (n)],[-\Psi (n),0]$ are called the zones of the uniform local normal attraction (z. u. l. n. a.) if $$\frac{p_{Z_n}(x)}{\frac1{\sqrt{2\pi}}e^{-x^2/2}}\to1$$ uniformly in x belonging to the said zones. Let $\alpha1/2$. The condition $$\mathbf E\exp\left|{x_j}\right|^{4\alpha/(2\alpha+1)}\infty$$ is proved to be necessary for the zones $[0,n^\alpha],[-n^\alpha,0]$, to be z. n. a., and for $x_j\in(d)$ to be the z. u. l. n. a. Let $\rho(n)$ be a given monotonic function increasing as slowly as we please, then the condition $(*)$ is sufficient for the zones $[0,n^\alpha/\rho(n)];[-n^\alpha/\rho(n),0]$ to be the z. n. a., and for $x_j\in(d)$ to be the z. u. l. n. a. if $\alpha1/6$. If $\alpha>1/6$, $x_j\in(d)$, a condition is given in terms of the series $1/6,1/4,3/10,\dots,(1/2)(s+1)/(s+3)\to1/2$ and of moments of $x_j$. This condition is necessary for the zones $[0,n^\alpha \rho (n)]$, $[-n^\alpha\rho(n),0]$ to be z. u. l. n. a. and sufficient for the zones $[0,n^\alpha/\rho (n)]$; $[-n^\alpha\rho (n),0]$ to be z. u. l. n. a.