Convergence of random sums in non-commutative probability

Random sums of independent random variables have been extensively studied in classical probability theory. We consider random sums of self-adjoint variables from a non-commutative probability space, and establish several $\ast$-convergence results. In particular, we show that the joint $\ast$-convergence of the standardized random sum of identically distributed self-adjoint variables and the standardized stopping random variable (rv) is equivalent to the convergence of all moments of the stopping rv together with the convergence of the ratio of its mean to its variance. We obtain central limit theorems for the random sums of free, independent and half independent self-adjoint variables with both deterministic and random scaling. Furthermore, we derive some scaling $\ast$-convergence limits for randomly indexed self-adjoint variables.