Strong law and central limit theorem for a process between maxima and sums

We prove an invariance principle for the random process (X n ) n1 given by where (Y n ) n1 are i.i.d. random variables and ( n ) n are nonrandom numbers tending upward to 1 (both in ). This process interpolates between maxima ( n 0) and sums ( n 1). Depending on the distribution ofY n and on the rate at which n 1 the scaling behaviour exhibits different regimes. Our techniques are flexible and are applicable to more general types of iterative schemes. Fulltext Previe