Extremal Processes, II

Abstract

Introduction.An extremal process is defined to be a stochastic process, {Y(t), _> 0}, having the following property" Y(tl) Y(t.) Y(tn) has the same joint distribution law as does U1 ;max (U1, U2);;max (U, U2, Un), (1.1) where 0 _< < < tn, n is arbitrary, U,..., U are inde- pendent random variables, andNotice the formal similarity between this definition of an extremal proc- ess and the definition of an i.d., stationary increment process.If "max (U1, U)" is replaced by "U1 + + U" the definition of an extremal process becomes the definition of an i.d.process.The motivating examples of extremal processes are studied in [2].These arise in connection with limiting distributions of maxima of independent and identically distributed random variables.Another example of an extremM process is the following.Let {Z(t), _< 0} be an i.d.stationary increment process.Define Y(t) maximum positive discontinuity of Z(u), 0 _< u _< t, with the understanding that this has value 0 if there are no positive discontinuities.It is easy to see that Y(t) is an ex- tremal process.The purpose of this paper is to describe the general structure of extremal processes, and to generalize and unify the results of [2].In Section 7 and thereafter extensions are made to multivariate processes.We also refer the reader to a paper by Lamperti [5], where the same class of processes is studied from a somewhat different point of view. The function Q.There is no loss in generality in supposing that Y(t) is a separable process and is non-decreasing in with probability 1, [1].(In fact, Theorem 4.1 describes an explicit representation of the process having these properties.)Hence, for any fixed u, P(Y(t) < u) is monotone in t.The definition of an extremal process implies that P(Y(h + t.) < u) P(Y(h) < u)P(Y(t) < u)for all h, t.Hence, there exists a constant Q Q(u) such that (2.1) P(Y(t) < u) exp-tQ(u).

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