Weak convergence with random indices

Suppose {Xnn[greater-or-equal, slanted]-0} are random variables such that for normalizing constants an>0, bn, n[greater-or-equal, slanted]0 we have Yn(·)=(X[n, ·]-bn/an => Y(·) in D(0.[infinity]) . Then an and bn must in specific ways and the process Y possesses a scaling property. If {Nn} are positive integer valued random variables we discuss when YNn --> Y and Y'n=(X[Nn]-bn)/an => Y'. Results given subsume random index limit theorems for convergence to Brownian motion, stable processes and extremal processes.