Product Representations for Random Variables with Weibull Distributions and Their Applications*

Abstract

In this paper, product representations are obtained for random variables with theWeibull distribution in terms of random variables with normal, exponential and stable distributions yielding scale mixture representations for the corresponding distributions. Main attention is paid to the case where the shape parameter γ of theWeibull distribution belongs to the interval (0, 1]. The case of small values of γ is of special interest, since the Weibull distributions with such parameters occupy an intermediate position between distributions with exponentially decreasing tails (e.g., exponential and gamma-distributions) and heavy-tailed distributions with Zipf–Pareto power-type decrease of tails. As a by-product result of the representation of the Weibull distribution with γ ∈ (0, 1) in the form of a mixed exponential distribution, the explicit representation of the moments of symmetric or one-sided strictly stable distributions are obtained. It is demonstrated that if γ ∈ (0, 1], then the Weibull distribution is a mixed half-normal law, and hence, it can be limiting for maximal random sums of independent random variables with finite variances. It is also demonstrated that the symmetric two-sided Weibull distribution with γ ∈ (0, 1] is a scale mixture of normal laws. Necessary and sufficient conditions are proved for the convergence of the distributions of extremal random sums of independent random variables with finite variances and of the distributions of the absolute values of these random sums to the Weibull distribution as well as of those of random sums themselves to the symmetric two-sided Weibull distribution. These results can serve as theoretical grounds for the application of the Weibull distribution as an asymptotic approximation for statistical regularities observed in the scheme of stopped random walks used, say, to describe the evolution of stock prices and financial indexes. Also, necessary and sufficient conditions are proved for the convergence of the distributions of more general regular statistics constructed from samples with random sizes to the symmetric two-sided Weibull distribution.