If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. Does anyone know an example of two different probability distributions with identical moments? The less pathological the better.
**Edit:** Is it unconditionally true if I specialize to discrete distributions?


And a related question: Suppose I ask the same question about Renyi entropies. Recall that the Renyi entropy is defined for all $a \geq 0$ by


$$ H\_a(p) = \frac{\log \left( \sum\_j p\_j^a \right)}{1-a} $$


You can define $a = 0, 1, \infty$ by taking suitable limits of this formula. Are two distributions with identical Renyi entropies (for all values of the parameter $a$) actually equal? How "rigid" is this result? If I allow two Renyi entropies of distributions $p$ and $q$ to differ by at most some small $\epsilon$ independent of $a$, then can I put an upper bound on, say, $||p-q||\_1$ in terms of $\epsilon$? What can be said in the case of discrete distributions?



When are probability distributions completely determined by their moments?
Roughly speaking, if the sequence of moments doesn't grow too quickly, then the distribution is determined by its moments. One sufficient condition is that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments. See Billingsley, *Probability and Measure*, chapter 30.


A standard example of two distinct distributions with the same moment is based on the lognormal distribution:


$$f\_0(x) = \sqrt{2 \pi} x^{-1} e^{-(\log x)^2 /2}$$


which is the density of the lognormal, and the perturbed version


$$f\_a(x) = f\_0(x)(1 + a \sin(2 \pi \log x))$$


These have the same moments; namely the nth moment of each of these is exp(n2/2).


A condition for a distribution over the reals to be determined by its moments is that lim supk → ∞ (μ2k)1/2k/2k is finite, where μ2k is the (2k)th moment of the distribution. For a distribution supported on the positive reals, lim supk → ∞ (μk)1/2k/2k being finite suffices.


This example is from Rick Durrett, *Probability: Theory and Examples*, 3rd edition, pp. 106-107; as the original source for the lognormal Durrett cites C. C. Heyde (1963) On a property of the lognormal distribution, J. Royal. Stat. Soc. B. 29, 392-393.


