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Suppose I have a probability distribution $A$ with continuous support over $\mathbb{R}$. Suppose $A$ has a sequence of finite (central) moments $\mu_1, \mu_2,\ldots,\mu_n$. I understand that $\mu_1$ is the mean, and $\mu_2$, $\mu_3$ and $\mu_4$ define variance, skewness, and kurtosis of the distribution, respectively.

I am wondering about the meaning of $\mu_5, \mu_6, \mu_7,\ldots$ When they are finite, what do they represent about the distribution $A$?

I understand that the odd central moments of the symmetric distribution are zero, so I am assuming that odd moments are related to the skew. What do even higher moments represent? I am particularly curious about $\mu_6$.

Also, suppose all moments of $A$ are finite. What does that say about $A$? Does it mean that $A$ has a specific representation? I've heard somewhere that all finite moments of $A$ with support $\mathbb{R}$ means that the tails of $A$ decay exponentially. Is that true? If so, can someone point me to a proof?

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...the odd central moments of the symmetric distribution are zero if they exist. :) – cardinal Nov 10 '11 at 0:26
up vote 2 down vote accepted

For your final statement the classic distribution to consider is the log-normal distribution. It is an example of a case where all moments are finite, but the moment generating function does not exist and the moments do not determine the distribution.

Take for example the log-normal distribution with parameters $\mu=0$ and $\sigma = 1/\sqrt2$ which has a distribution on positive values with density $$f(x)=\frac{1}{ x^{1+\log_e x}\sqrt{\pi} }$$

This has finite moments about $0$ of $E[X^n]= \exp(n^2/4)$ but the density of the tail does not decay exponentially.

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The moment generating function does not exist in a neighborhood around zero, which is what is key. It does, of course, exist, for $t \leq 0$ since the lognormal takes on only nonnegative values. – cardinal Nov 10 '11 at 0:32
I see. But if MGF exists around 0, then, by Chernoff bound, the tails decay exponentially (this is my interpretation of Lemma 11.9.1 in Cover and Thomas's "Elements of Information Theory" 2nd edition). And if MGF exists around 0, all moments exist and are finite. Are those two statements correct? – M.B.M. Nov 10 '11 at 0:58

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