 Gaussian smoothing of an integer-valued distribution

Let $X$ take integer values with probabilities $p_n$, and let $Y_{\sigma }$ be an independent centered normal distribution with positive standard deviation $\sigma$. Conditioning on $X$ gives the probability density function of their sum as $g_{\sigma }(z)={\sum }_n{p}_n{\phi }_{\sigma }(z-n)$, where ${\phi }_{\sigma }$ is the normal density. Nonnegativity and Tonelli's theorem show that this density integrates to one. For small $\sigma$, isolated atoms become narrow smooth peaks with their original probability masses.

For each fixed real $t$, dominated convergence givessince $X/\sigma \to 0$ almost surely and ${\phi }_1$ is bounded. Thus every fixed integer-valued distribution acquires a Gaussian central profile on this expanding scale, without any moment assumption on $X$. This is not a claim about uniform relative error in the tails or about finite moments. For example, if $p_n=3/({\pi }^2{n}^2)$ for $n\ne 0$ and $p_0=0$, the expectation of $|X|$ is infinite. Since $\mathbb{E}|Y_{\sigma }|<\infty$, the triangle inequality forces $\mathbb{E}|X+Y_{\sigma }|=\infty$ too. Symmetry then gives equal infinite positive and negative first moments, so there is no ordinary mean. The second moment is also infinite. A Gaussian-looking center can therefore coexist with a heavy-tailed distribution.