Probability Distributions: Moment Analysis

We deal with random variables and their distributions, discrete or continuous, assuming their all moments are finite. Such a distribution is either uniquely determined by its moments ($M$-determinate), or it is non-unique ($M$-indeterminate). We focus on recent developments and give checkable conditions allowing to decide if a distribution is $M$-determinate or $M$-indeterminate. We analyze nonlinear Box-Cox transformations of random data and their M-determinacy. New results will be reported to cover distributions of random variables, vectors and the solutions of SDEs. The M-determinacy is important for both theory and applications in other areas, including Financial Mathematics.