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On the moment determinacy of products of non-identically distributed random variables

Lin G. D., Stoyanov J.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 1

21--33

We show first that there are intrinsic relationships among different conditions, old and recent, which lead to some general statements in both the Stieltjes and the Hamburger moment problems. Then we describe checkable conditions and prove new results about the moment (in)determinacy for products of independent and non-identically distributed random variables. We treat all three cases: when the random variables are nonnegative (Stieltjes case), when they take values in the whole real line (Hamburger case), and the mixed case. As an illustration we characterize the moment determinacy of products of random variables whose distributions are generalized gamma or double generalized gamma all with distinct shape parameters. Among other corollaries, the product of two independent random variables, one exponential and one inverse Gaussian, is moment determinate, while the product is moment indeterminate for the cases: one exponential and one normal, one chi-square and one normal, and one inverse Gaussian and one normal.

On the geometric compounding model with applications

Hu C. Y., Lin G. D.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

135--147

Under the geometric compounding model, we first investigate the relationship between the compound geometric distribution and the underlying distribution, including the preservation of the infinite divisibility property. An interesting upper bound for the tail probability of the compound geometric distribution is provided by using only the mean of the underlying distribution. Secondly, we apply the obtained results to understand better the L-class of life distributions. In particular, we strengthen a surprising result of Bhattacharjee and Sengupta [5] and show that there are life distributions FєL with the following properties: (i) the support of F consists of countably infinite points, (ii) (ii) the coefficient of variation of F is equal to one, and (iii) (iii) F is not in the HNBUE class (the harmonic new better than used in expectation class).Finally, we apply geometric compounds to characterize the semi-Mittag-Leffler distribution and extend a known result about the exponential distribution.