Normality of sum of independent random variables implies normality of each

This theorem is stated as a footnote in a lecture:

Let $X,Y$ be independent random variables

1. If $X+Y$ follows a normal distribution, then $X$ and $Y$ follow a normal distribution.

2. If $X+Y$ follows a Poisson distribution, then $X$ and $Y$ follow a Poisson distribution.

Note that $X$ and $Y$ are not supposed to be identically distributed. $X$ may follow Poisson(1) while $Y$ follows Poisson(2).

I've been trying to prove these statements without success. Sums of independent random variables hint at something involving characteristic functions (inverse Fourier transform or some such), but I've been unsuccessful so far.

• Can you provide a link to the mentioned lecture? My original belief was that 1. does not hold, but I am having a hard time in contructing a counter-example which meets Bochner's constraints (en.wikipedia.org/wiki/Bochner%27s_theorem), so I am starting to think that such constraints can be met only if both $X$ and $Y$ are normally distributed. – Jack D'Aurizio Nov 1 '17 at 22:07
• @JackD'Aurizio it seems to be a famous theorem by Cramer, see here – Gabriel Romon Nov 1 '17 at 22:13
• Indeed, a proof of 1. is hidden at page 443 here – Jack D'Aurizio Nov 1 '17 at 22:22
• And a proof by entropy has been proposed by Per Vognsen on MO – Jack D'Aurizio Nov 1 '17 at 22:29
• The book by Stoyanov called "Counterexamples in probability" or a similar title, has these and tons of other similar results. – Did Nov 1 '17 at 23:17

The moment generating function of a Poisson distribution with parameter $\lambda$ is given by $\exp\left(\lambda(e^t-1)\right)$. In general, if $f(t)$ is the MGF of a distribution supported on $\mathbb{R}_{\geq 0}$, the sequence given by $\{f^{(n)}(0)\}_{n\geq 0}$ is non-negative and log-convex by the Cauchy-Schwarz/Holder inequality. The Poisson distribution has the remarkable property of having constant cumulants. If we consider the cumulant generating functions $K_X(t)$ and $K_Y(t)$ and assume that $\kappa_n=K_X^{(n)}(0)$ is non-constant, we should reach a contradiction by exploiting log-convexity and the inverse Laplace transform.