Mellin-Stieltjes Transforms in Probability Theory

Mellin-Stieltjes transforms are very useful in solving problems in which products and ratios of random variables are encountered. The paper relates some general considerations pertaining to the application of these transforms (Section 1), and also gives a concrete example of their use in studying analytical properties of stable distributions (Section 2). In the first section the relationship between the Mellin-Stieltjes transform, the unilateral Laplace-Stieltjes transform and the characteristic function of a given distribution is established. For the sake of simplicity, all distributions in Section 1 are considered as being continuous at zero. The concepts “truncation” and equivalence of random variables are also introduced there. Any random variable having a distribution function \[ \tilde F(x) = \frac{{F(x) - F(0)}}{{1 - F(0)}},\ x \geqq 0;\quad \tilde F(x) \equiv 0,\ x \leqq 0 \] is called the truncation$\xi $of the random variable$\xi $having a distribution function$F(x)$. Truncations may also-be considered as functions of an initial random variable $\xi $ (two different representations of $\xi $ are given as a function of $\xi $). Random variables $\xi _1 $ and $\xi _2 $ are considered as being equivalent and are designated as $\xi _1 \approx \xi _2 $ if the distribution functions corresponding to them are equal. The concepts of truncations and equivalency of random variables are systematically employed in the second section in establishing precise and limiting relationships in the class of stable distributions. Stable distributions naturally decompose into two analytically independent branches if the shift parameter $\gamma $ is specially selected. For $\alpha \ne 1$ it is possible to determine an explicit expression of the Mellin transform for these branches employing Euler’s $\Gamma $-function. These two circumstances justify the use of the above mentioned concepts. This explicit representation of the Mellin transforms in turn is used to determine a whole series of relationships between branches of stable distributions, in which all previously known relationships of the same type are special cases. It should be noted that in paragraph 2.6 all random variables, which are written separately, are considered independent. The behaviour of stable distributions near critical points $\alpha = 0$ and $\alpha = 1$ is investigated in the second section.