Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete entropy
\begin{equation}
H(\lfloor X \rfloor) = - \sum\_{n\in\mathbb{Z}} \mathbb{P}( \lfloor X \rfloor = n ) \log( \mathbb{P}( \lfloor X \rfloor = n ) ), 
\end{equation}
which is in $[0,\infty]$ as a sum of nonnegative terms, since $- x \log x \geq 0$ for any $0 \leq x \leq 1$ (with the convention $0\log 0 = 0$).


I am looking for sufficient conditions such that $H(\lfloor X \rfloor ) < \infty$. For instance, is it sufficient to know that $X$ has a finite absolute moment $\mathbb{E}[|X|^p] < \infty$ for some $p>0$? Any condition of this type, possibly weaker, is welcome.


*Motivation:* There are strong connection between the differential entropy of $X$ (assuming $X$ has a pdf whose differential entropy is well-defined) and the discrete entropy of $\lfloor nX \rfloor$ when $n\rightarrow0$. This was the main topic of the 1959 paper from Alfred Rényi intitled *On the dimension and entropy of probability distributions*: I am questioning the assumptions under which the discrete entropy is well-defined.



Discrete entropy of the integer part of a random variable
Since $\lfloor X\rfloor$ has finite entropy if and only if $|\lfloor X\rfloor|$ has finite entropy, it suffices to consider random variables taking values in the natural numbers. Write $p\_n$ for $\mathbb P(X=n)$ (so that $\sum\_n p\_n=1$). We have $X\in L^q$ if and only if $\sum p\_n n^q<\infty$.


Suppose $X\in L^q$ so that $\sum p\_n n^q<\infty$.
Then let $S\_1=\{n\colon p\_n<\frac{1}{n^2}\}$ and $S\_2=\{n\colon p\_n\ge \frac 1{n^2}\}$.
We have
$$
H(X)=\sum\_n -p\_n\log p\_n=-\sum\_{n\in S\_1}p\_n\log p\_n-\sum\_{n\in S\_2}p\_n\log p\_n.
$$
Since $-t\log t$ is increasing on $[0,\frac 1e]$, the first sum is bounded above by
$$
\sum\_{n\in S\_1}\frac{2\log n}{n^2}<\infty.
$$
There exists an $n\_0$ so that for $n\ge n\_0$, $2\log n<n^q$. For $n\in S\_2$ such that $n\ge n\_0$, $-\log p\_n<2\log n<n^q$, so that
$$
-\sum\_{n\in S\_2,\,n\ge n\_0}p\_n\log p\_n\le 
\sum\_{n\in S\_2,\,n\ge n\_0}p\_n n^q<\infty.
$$
Hence $H(X)<\infty$. (This trick appears in a couple of papers of mine: one with Ciprian Demeter in NYJM and another more recent preprint with Tamara Kucherenko and Christian Wolf).


