Three problems involving moments determinacy of distributions

If a distribution, say F, has all moments finite, then either F is unique (M-determinate) in the sense that F is the only distribution with these moments, or F is non-unique (M-indeterminate). In the latter case we can construct a Stieltjes class consisting of infinitely many distributions all having the same moments as F. There are classical and new criteria for uniqueness and for non-uniqueness. We make use of them when dealing with problems in the following areas:

• Box-Cox transformations of random data.
• Random sums of random variables.
• Identifiability of mixture distributions.

There is a rigorous and beautiful theory behind all this, but also there are important implications for statistical practice. If time permits, open questions will be outlined.
The material will be presented in an [hopefully] attractive way and the speaker will appeal to the audience for prompt comments. The talk will be addressed to both professionals in statistics and/or probability and graduate students in these areas.