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On the distribution of the maximum of successive cumulative sums of independently but not identically distributed chance variables

1948; American Mathematical Society; Volume: 54; Issue: 4 Linguagem: Inglês

10.1090/s0002-9904-1948-09021-8

1088-9485

Abraham Wald,

Probability and Statistical Research

The distribution of MN, in particular the limiting distribution of a suitably normalized form of MN, has been studied by Erdos and Kac [ l ] 1 and by the author [2] in the special case when the X's are independently distributed with identical distributions. In this note we shall be concerned with the distribution of MN when the X's are independent but not necessarily identically distributed. In particular, the mean and variance of Xi may be any functions of i. In §2 lower and upper limits for MN are obtained which yield particularly simple limits for the distribution of MN when the X's are symmetrically distributed around zero. In §3 the special case is considered when Xi can take only the values 1 and — 1 but the probability pi that Xi = 1 may be any function of i. The exact probability distribution of MN for this case is derived and expressed as the first row of a product of N matrices. The limiting distribution of MN/N is treated in §4. Since the interesting limiting case arises when the mean of Xi (i^N) is not only a function of i but also a function of N, we have to introduce a double sequence of chance variables. That is, for any N we consider a sequence of N chance variables XNU • • • f XNNLet pNi denote the mean and SNNWith the help of a method used by Erdos and Kac [ l ] , the following theorem is established in §4 :