Finite entropy and finite moments

This question is about when a discrete distribution has finite entropy because it has finite moments.

Let $$X$$ be a discrete random variable (which can be positive, negative, and/or zero unless stated otherwise), and let— $$H(X) = \sum_n -\mathbb{P}[X=n] \ln(\mathbb{P}[X=n]),$$ be the Shannon entropy of $$X$$.

It is known that:

• If $$X$$ is integer-valued and has a finite variance, then $$H(X)$$ is finite (Massey 1988).
• If $$X$$ is integer-valued and 0 or greater, then $$H(X)$$ is finite (Rioul 2022).

It is also known that not all discrete distributions have a finite Shannon entropy (one example is some members of the zeta Dirichlet distribution); see also Devroye and Gravel 2020.

However, I believe that this is so because the infinite-entropy zeta Dirichlet distributions (I think) have an infinite $$z$$th moment for any real $$z>0$$.

Moreover, I believe that the Cauchy distribution has a finite entropy even though its mean is infinite because it does have a finite $$z$$th moment for some $$z$$ in $$(0, 1)$$, in fact for every $$z$$ in $$(0, 1)$$.

However, several questions remain on the relationship between finite entropy and finite moments.

Questions

• If $$X$$ is supported on the whole set of integers and is such that $$\mathbb{E}[X^2]$$ is infinite but $$\mathbb{E}[X^z]$$ is finite for some $$z$$ in $$[1, 2)$$, then is $$H(X)$$ finite? If not, under what additional conditions is $$H(X)$$ finite?
• If $$X$$ is supported on the integers or some subset thereof, such that $$\mathbb{E}[X]$$ is infinite but $$\mathbb{E}[X^z]$$ is finite for every $$z$$ in $$(0, 1)$$, then is $$H(X)$$ finite? If not, under what additional conditions is $$H(X)$$ finite?
• If $$X$$ is supported on the integers or some subset thereof, such that $$\mathbb{E}[X]$$ is infinite but $$\mathbb{E}[X^z]$$ is finite for some $$z$$ in $$(0, 1)$$, then is $$H(X)$$ finite? If not, under what additional conditions is $$H(X)$$ finite?
• If $$X$$ is supported on the integers or some subset thereof, such that $$\mathbb{E}[X^z]$$ is infinite for every real $$z > 0$$, then is $$H(X)$$ infinite?

Motivation

My motivation is to characterize the discrete distributions with finite entropy, since only finite-entropy distributions can be sampled in finite time on average (Knuth and Yao 1976).

References

I assume that $$X$$ takes non-negative integral values.

First, suppose that $$E X^a<\infty$$ for some $$a>0$$. Then for all $$n\ge 0$$, it holds either that $$p_n\log \frac{1}{p_n}\le n^a p_n$$ or $$p_n\le e^{-n^a}.$$ But in the latter case, we can also bound, say, $$\log(1/p_n)\le \sqrt{p_n}$$. So both upper bounds are summable, and we get $$H(X)<\infty$$.

Reciprocally, if we know that $$E X^a=\infty$$ for all $$a>0$$, there is not much we can say about $$H(X)$$. Consider for example $$p_n=1/m^2$$ if $$n=e^m$$ for some $$m\ge 0$$ and $$p_n=0$$ otherwise. You can check that $$p_n$$ satisfies the all-infinite-moments condition (in fact, the main term of the sum does not even converge to $$0$$), but $$\sum_{n\le e^m} p_n\log(1/p_n)=\sum_{k\le m} \frac{\log(k^2)}{k^2}.$$ So $$H(X)<\infty$$.

• Thank you for the response. Can you answer for the case when $X$ can be negative as well as positive? All along I intended the case when $X$ can be positive and/or negative, unless stated otherwise in the question, especially the case when $X$ can take any positive or negative integer. Feb 18 at 18:27
• @PeterO. - so what does the moment condition correspond to now? Should we put absolute values?
– md5
Feb 18 at 19:04
• Only $z$th moments for any real $z > 0$, and the absolute values of the moments are taken (e.g., where I said "$\mathbb{E}[X^z] = \infty$", I meant "$\mathbb{E}[X^z]$ is infinite" or "$\mathbb{E}[X^z]$ is positive infinity or negative infinity"). Feb 18 at 19:12
• @PeterO. - but this is not even defined if the exponent is not an integer
– md5
Feb 18 at 20:31
• In that case, only one question remains to be asked: If $X$ is supported on the whole set of integers and has a finite mean, then is $H(X)$ finite? Feb 18 at 20:43