The classical Marcinkiewicz theorem (1939) states that if a random variable $X$ has a Laplace transform/characteristic function of the form $\mathbb{E}(e^{tX})=e^{P(t)} $ with $P$ a polynomial, then deg($P(t)$)$\leqslant 2$. Since $t=0$ must give $1$, we have $P(t) = \alpha t^2 + \beta t$ for certain real $\alpha,\beta$.
Now, consider the following function, for $a \in\mathbb{R}_+ $ : $ \phi_a(0) = 0 $ and $ \phi_a(k) = a (k - 1)^2 $ for all integer $ k \geqslant 1 $. We know that $ x \mapsto e^{ a (x - 1)^2 } $ cannot be a Laplace transform since the value in $ x = 0$ does not give $1$, but can there be a random variable $ X_a $ such that $ \mathbb{E}(e^{k X_a}) = e^{ \phi_a(k) } $ for all $ k \geqslant 1 $ ? In this case, what would be $ \mathbb{E}(e^{t X_a}) $ for $ t \in \mathbb{R} $ ?
Other related question : can we find such an extrapolation of $ \phi_a $ from the integers to the reals under the form $ a x^2 + \int_{\mathbb{R} } (e^{x \theta} - 1 - x\theta) d\mu(\theta) $ for a certain Levy measure $ \mu $ ? Namely, if $ X_a $ exists, is it infinitely divisible ?
