Triviality of the Limit Distributions Class for Sums of Random Numbers of Positive Random Variables

We prove that any positive random variable $Y$ represents a limit $Y=\lim_{m\to\infty}\sum_{k=1}^{\nu_m}X_k$ of a suitably chosen sequence of random sums $\{\sum_{k=1}^{\nu_m}X_k, m\geq 1\}$ of independent identically distributed positive random variables $X_k$ with a finite mean. For this to be true, the probability generating function $\mathcal{P}_m(z)$ of integer-valued non-negative random variable $\nu_m$ must be expressible as $\mathcal{P}_m(z)=\psi(m(1-z))$, where $\psi(s)$ is a Laplace transform of a probability distribution on the positive semiaxis.