Ergodic theorems

Abstract

We are now ready to resume the study of S _n . The first term Y′(n) in the dissection formula (14.3) is simple; the second is the sum of a random number l(n) — 1 of independent, identically distributed random variables Y _v . There is dependence between l(n) and the Y _v that has to be circumvented or analyzed. The third term Y″(n), treated in Theorem 14.8, may at first sight appear quite similar to the first, but in reality it is somewhat more troublesome due to the greater dependence on n. In spite of these foreseen difficulties it is expected that the classical limit theorems for sums of independent identically distributed random variables should apply to S _n , under suitable conditions. We proceed to investigate this. The following notation will be used in the sequel:

• Plim = limit with probability one;

• plim = limit in probability;

similarly for $P\underline{\lim }$ and $P\overline{\lim }$.