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- W2079208053 abstract "In § 1 limit theorems for sums of independent random variables are considered in which the limit distribution differs from the normal law. Theorem 1. Let${ xi _i } $be a sequence of independent random variables which have the following distribution functions[ F_i (x)left{ begin{gathered} 0,quad x leqq 0 hfill 1 - left[ {c_i + alpha _i (x)} right]frac{1} {{x^alpha }},quad x > 0, hfill end{gathered} right. ]where$left| {alpha _i (x)} right| < alpha (x)$$lim _{x to infty } alpha (x) = 0,0 < c' leqq c_i leqq c'' < infty ,i = 1,2, cdots $and$0 leqq alpha leqq 1$. Then[ mathop {lim }limits_{n to infty } Pleft{ {frac{{xi _1 + cdots + xi _n }} {{B_n }} < x} right} = F_alpha (x), ]where$B_n left( {sum _{i = 1}^n c_i } right)^{{1 / alpha }} $and the distribution function$F_alpha (x)$has a characteristic function[ f_alpha (t) = exp left{ { - Gamma (1 - alpha )cos frac{{pi alpha }} {2}|t|alpha left[ {1 - ifrac{t} {{|t|}}tgfrac{{pi alpha }}{2}} right]} right}. ] The main result in § 2 is as follows: Theorem 4. Let${ F_1 (x)} $be a set of absolutely continuous distribution functions and let the following conditions be fulfilled: $1^0 |p_1 (x)| < c < infty $where$F_1 (x) = int_{ - infty }^x {p_1 (y)dy} $andcis one and the same constant for all$F_1 (x)in {F_1 (x)}$, $2^0 |F_1 ( - x) + 1 - F_1 (x)| < C(x),x > 0$where$C(x)$is one and the same function for all$F_1 (x) in { F_1 (x)} $. If$|f_1 (t) - f_2 (t)| < varepsilon $, $|t| < T$where$f_1 (t)$is a characteristic function of$F_1 (t) in { F_1 (t)} $and$f_2 (t)$is a characteristic function of some distribution function$F_2 (x)$, then[ mathop {sup }limits_x left| {F_1 (x) - F_2 (x)} right| < N(beta )left[ {varepsilon frac{{16c}} {{pi beta T}}} right] + 3beta ]for any $beta $, $0 < beta < 1$where$N(beta )$is a function depending on${ F_1 (x)} $." @default.
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- W2079208053 date "1958-01-01" @default.
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- W2079208053 title "Some Problems in the Field of Limit Theorems" @default.
- W2079208053 doi "https://doi.org/10.1137/1103016" @default.
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