We deduce the non-asymptotical bilateral estimates for moment inequalities for sums of non-negative independent random variables, based on the correspondent estimates for the so-called Bell functions and the Poisson distribution.

We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between the tail behavior, moments (Grand Lebesgue Spaces) norm and Orlicz’s norm for random variables (r.v.), which does not satisfy in general case the Kramer’s condition.

We construct a Banach rearrangement invariant norm on the measurable space for which the finiteness of this norm for measurable function (random variable) is equivalent to suitable tail (heavy tail and light tail) behavior.We investigate also a conjugate to offered spaces and obtain some embedding theorems.Possible applications: Functional Analysis (for instance, interpolation of operators), Integral Equations, Probability Theory and Statistics (tail estimations for random variables) etc.

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