The Calculation of the Density and Distribution Functions of Strictly Stable Laws

Abstract

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters ( α , θ ) . A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter λ .

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

“…The proof of this lemma can be found in the paper [3]. To obtain the probability density, there is no fundamental difference which of the formulas to use on the right side (2).…”

“…The latter distribution first came out in the book by V.M. Zolotarev [1] (see formula (2.3.5a)) and later was examined in the works [2,3]. Different representations for stable laws are required to calculate the probability density or distribution function in other cases.…”

“…The second way of obtaining integral representations is the application of the stationary phase method when calculating the integral in (2) (see [1,3,15,16]). The advantage of this method of inverting the characteristic function is that the resulting integral representation is expressed in terms of a definite integral of a monotonic function.…”

“…The advantage of this method of inverting the characteristic function is that the resulting integral representation is expressed in terms of a definite integral of a monotonic function. Such integral representations were obtained for stable laws with different parameterizations of the characteristic function: for parameterization "B" in the works [1,15], for parameterization "M" in the paper [16], for parameterization "C" in the paper [3]. (Here, the notation of various parameterizations of the characteristic function is given in accordance with the designations introduced in the book by V.M.…”

“…Obtaining such expansions turns out to be necessary in connection with the problem of calculating the probability density and distribution function of stable and fractionally stable laws. In fact, in the article [3] integral representations were obtained for the probability density and distribution function of a strictly stable law with the characteristic function (1). Since these integral representations were obtained using the stationary phase method, then with small and large values of the coordinate x the integrand has the form of a very sharp peak.…”