Limits of Probability Measures with General Coefficients

Abstract

We study the convergence of probability measures in terms of moments by applying operators to their Bessel generating functions. We consider a general setting of applying operators such as the Dunkl operator to formal power series that are symmetric or symmetric in all but one variable. Afterwards, we apply the results from this setting by considering Bessel generating functions as the formal power series to obtain a Law of Large Numbers as $N$, the number of variables, increases to infinity and $N^c\beta$ converges to a constant, where $c\in (-\infty, 1)$. In contrast with previous results, we consider when the scaled partial derivatives of the logarithms of the Bessel generating functions evaluated at the origin can have nonzero $N\rightarrow\infty$ limit when any number of variables is involved. Then, the free cumulant of order $k$ is a linear combination of the limits of the order $k$ partial derivatives.