A note on random sums with nonrandom centering (Q1368992)

Let \(\{X_n\}\) be a sequence of i.i.d. random variables and \(\{N_n\}\) a sequence of nonnegative integer valued random variables independent of \(\{X_n\}\). Assume that \((X_1+\cdots+X_n)/n\to\mu\not=0\) in probability, \(\gamma_n\to\infty\), and \(\gamma_n/n\to 0\). Put \[ R_n=\frac{ X_1+\cdots+X_{N_n}-n\mu }{\gamma_n},\qquad T_n=\frac{ X_1+\cdots+X_n-n\mu }{\gamma_n}, \qquad U_n=\frac{ N_n-n }{\gamma_n}\mu. \] The author studies tightness and weak convergence of random sums \(\{R_n\}\) with nonrandom centering. One of the results is the following Theorem 3. If any two of \(R_n\), \(T_n\), or \(U_n\) converge in distribution, then so does the third. Furthermore if \(R_n\to R\), \(T_n\to T\), and \(U_n\to U\) in distribution, then \(R=T+U\) in distribution where \(T\) and \(U\) are independent. Random sums of the form \(R_n\) have been studied recently, in paticular by \textit{M. Finkelstein} and \textit{H. G. Tucker} [Proc. Am. Math. Soc. 107, No. 4, 1061-1070 (1989; Zbl 0682.60017)] and \textit{M. Finkelstein, V. M. Kruglov} and \textit{H. G. Tucker} [J. Theor. Probab. 7, No. 3, 565-598 (1994; Zbl 0806.60012)].