The calculation of the probability density of a strictly stable law at large $X$

Abstract

The article is devoted to the problem of calculating the probability density of a strictly stable law at x → ∞. To solve this problem, it was proposed to use the expansion of the probability density in a power series. A representation of the probability density in the form of a power series and an estimate for the remainder term was obtained. This power series is convergent in the case 0 < α < 1 and asymptotic at x → ∞ in the case 1 < α < 2. The case α = 1 was considered separately. It was shown that in the case α = 1 the obtained power series was convergent for any |x| > 1 at N → ∞. It was also shown that in this case it was convergent to the density of g(x, 1, θ). An estimate of the threshold coordinate x N ε , was obtained which determines the range of applicability of the resulting expansion of the probability density in a power series. It was shown that in the domain |x| x N ε this power series could be used to calculate the probability density.

The Calculation of the Probability Density and Distribution Function of a Strictly Stable Law in the Vicinity of Zero

“…Now we examine the issue of the convergence of the obtained expansion. Since this expansion was obtained by integrating the expansion for the probability density, taking into account the results of Corollary 1, proved in the paper [16], one can state that this series converges in the case of α < 1 for all x, in the case α = 1, only for |x| > 1, and in the case α > 1 the series is asymptotic one at x → ∞. A more precise formulation is given by the following corollary.…”

“…To solve the stated problem, we need to expand the probability density into a series at x → ∞. A similar expansion was obtained in the article [16], where the following theorem was proved.…”

“…where g ∞ N (x, α, θ) is the N -th partial sum and R ∞ N (x, α, θ) is the remainder term of the series. In the case x → ∞ the first summand is determined by the expression (3) and for the remainder term, we use the expression obtained in the article [16]…”

“…In particular, the integration variable in the formula (9). See the paper [16] for detail. Hence, here, and further in the text, we will assume everywhere that τ /x → 0.…”

“…The most suitable idea is to use expansions of the probability density and distribution function in power series at x → 0 and x → ∞. The articles [15,16] show that the use of expansions of stable laws in power series at x → 0 and x → ∞ makes it possible to solve the problem of calculating stable laws completely at very small and very large x. However, in these articles the problem of calculating the distribution function of a strictly stable law in the case of x → ∞ was left out of consideration.…”