Transformations of random data: determinacy of the distributions

Functional transformations of random data (usually called Box-Cox transformations) are intensively studied and widely used in statistical practice. Data may come from observations of random variables or of stochastic processes.
Our goal is to analyse the distributions of the data, before and/or after transforming, and their properties expressed in terms of the moments.
We concentrate on the classical problem of moments (originated by Chebyshev, Markov and Stieltjes). It is more than curious to know conditions under which a distribution is uniquely determined by its moments and other cases when the distribution is non-unique. And, what about the determinacy when transforming the data? After a brief survey of available criteria and conditions (e.g. Carleman, Hamburger, Cramer, Hausdorff), we turn to some very recent developments. All statements and criteria will be well illustrated by examples based on popular distributions such as Normal, Exponential, Log-normal, Gamma, Poisson, IG, etc. Several facts will be reported, some of them not so well-known, and even surprising.
There is a serious and beautiful theory behind all this, including open questions. However, there are important implications for the applications. The material will be presented in an understandable and attractive way, addressing the talk not only to professionals in statistics, probability, mathematics but also to doctoral and master students in these areas.