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

The independent identically distributed variables X₁, X₂, , Xₙ are supposed to have E (Xⱼ) = 0;D (Xⱼ) = ² x) }}{1{{ {2 }}ₓ^ e^{ - n² /2dn } }} 1 \] for x 0, (n) ; the zones - (n), 0 are defined similarly as z. n. a. . The zones 0, n^, - n^, 0 (> 0 constant) are called simplest. The zones such that (n) = o (n^1/ 6) are called “narrow”. For the random variables of the class (d) (possessing a bounded continuous density) the zones 0, (n), - (n), 0 are called the zones of the uniform local normal attraction (z. u. l. n. a. ) if pZₙ (x) {1{{ {2 }}e^ - x² / 2 }} 1 uniformly in x belonging to the said zones. Let 1/ 6, Xⱼ (d), a condition is given in terms of the series, 1/ 6, 1/4, 3/10, , (1/2) (s + 1) / (s + 3) 1/ 2 and of moments of Xⱼ. This condition is necessary for the zones 0, n^ (n), - n^ (n), 0 to be z. u. l. n. a. and sufficient for the zones 0, n^/ (n), - n^ (n), 0 to be z. u. l. n. a. .