Abstract
We use the composition method to analyse the convolutions of probability distributions by emploing the Appell polynomials and Bergström identity. Our approximation is based on the probability distributions which have the inverse generalized measure of bounded variation. The idea to use the accompanying probability distribution eλ(F−E), λ > 0, was first proposed by B.V. Gnedenko [1].
Highlights
We consider the convolutions of one-dimensional distribution Fn1, Fn2, . . . , Fnkn defined by the formula knFn = ∗ Fnj = Fn1 ∗ · · · ∗ Fnkn j=1 and analyse the asymptotic behavior of Fn, when n → ∞ and kn → ∞
Has the inverse generalized measure F − of bounded variation, i.e., F ∗F − = F −∗F = E, where E is the degenerate at the point 0 probability distribution, to evaluate the difference
In (11), we present a new concept of the generalized Appell polynomials
Summary
We consider the convolutions of one-dimensional distribution Fn1, Fn2, . . . , Fnkn defined by the formula kn. Lemma 1 contains the estimate of the sum of the coefficients of Appell’s polynomials. This new inequality is used in Theorem 3 to estimate the remainders of the expansion of the convolutions in Appell’s polynomials. In (11), we present a new concept of the generalized Appell polynomials. In Theorem 4, we expand the convolution through the generalized Appell polynomials and find the convergence conditions. We intend to use this technique to analyze the convolutions of probability distributions defined on algebraic structures. We define the generalized Appell polynomial by j−1.
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