# “Physical” meaning of higher moments (their values and their existence)

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

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.
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