Limit theorem for the maximum of an exponential autoregressive process. Technical report No. 14

The asymptotic behavior of the maximum of a particular autoregressive process is discussed. The process was introduced by Gaver and Lewis in 1975 as a generalization of the Poisson process which allows for some dependence in the successive interarrival times. The exact distribution of the maximum of the first two terms and the first three terms in the sequence (denoted by M/sub 1/ and M/sub 2/, respectively) is calculated, and upper and lower bounds are obtained for M/sub n/, the maximum of the first n + 1 terms in the sequence. Loynes in 1965 gave conditions under which a stationary stochastic process has a maximum which behaves in the limit just as for independent identically distributed variables with the marginal distribution of the stationary process. Such conditions are often very difficult to verify in practice. However, for this particular example the joint distribution of the process at time n and n + j + 1 is determined, and the conditions to obtain the limit theorem are verified by use of the Markov property.