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 ) = σ ² < ∞. Denote \[ Z_n = {{X_1 + ⋯ + X_n }}{{σ √ n }}. \] Let ψ (n) → ∞ be some monotone function. The sequence of segments [0,ψ (n)] is called the zone of normal attraction (z. n. a.) if \[ {{{ P}(Z_n > x)}}{{{1}{{√2π }}∫_x^∞ {e- n^2 /2dn } }} → 1 \] for x ∈ [0,ψ (n)]; the zones [ - ψ (n),0] are defined similarly as z. n. a.. The zones [0,n^α ],[ - n^α ,0](α > 0 constant) are called simplest. The zones such that ψ (n) = o(n1/ 6 ) are called “narrow”. For the random variables of the class $(d)$ (possessing a bounded continuous density) the zones [0,ψ (n)],[ - ψ (n),0] are called the zones of the uniform local normal attraction (z. u. l. n. a.) if pZₙ (x) 1√2π e- x² / 2 → 1 uniformly in x belonging to the said zones. Letα < 1/2. The condition \[ { F}exp | {X_j } |4α /(2α + 1) < ∞ \] is proved to be necessary for the zones [0,n^α ],[ - n^α ,0], to be z. n. a., and for Xⱼ ∈ (d) to be the z. u. l. n. a.. Let ρ (n) be a given monotonic function increasing as slowly as we please, then the condition $( * )$ is sufficient for the zones [0,n^α/ρ (n)],[ - n^α/ρ (n),0] to be the z. n. a., and for Xⱼ ∈ (d) to be the z. u. l. n. a. if α < 1/ 6. If α > 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^α ρ (n)], [ - n^α ρ (n),0] to be z. u. l. n. a. and sufficient for the zones [0,n^α/ρ (n)], [ - n^αρ (n),0] to be z. u. l. n. a..