Studia Scientiarium Mathematicarum Hungarica 31. (1996)

1-3. szám - Auer P.-Hornik K.: Limit laws for the maximal and minimal increments of the Poisson process

2 P. AUER and K. HORNIK which denote the maximal and minimal number of points of t] in subsets of [0,T]d given by unions of L disjoint sets E{ from T2, such that all sets Ei have volume V1. (In what follows, a family {Ei} of sets is called “disjoint” if A(2?, n Ej) =0 for all i^j.) In the one-dimensional case, the natural choice for the family 2 is the set 2int of all intervals in [0,1]. Multi-dimensional generalizations are e.g. Í^UBi the set of axis-parallel cubes in [0, l]d, 2gPH, the set of spheres in [0, l]d, 2REC, the set of axis-parallel (generalized) rectangles in [0, l]d, 2[(CUB, the set of cubes in general position in [0, l]d, and 2^REC, the set of rectangles in general position in [0, l]d. For u ER, let h(u)u log u — u + 1, u0, oo u < 0, be the familiar Chernoff function of the Poisson distribution, and let for (f> > 0 u+{4>) = inf{w > 1 : h{u) ^ <j>], u~ {(j)) = sup{u < 1 : h[u) ^ cf>}. Finally, for all x € R, [x] and [x] denote the largest integer ^ x and the smallest integer > x, respectively. 2. First results Révész [14] was first to obtain the exact rate of Aj(l) by proving the following theorem. Theorem 2.1. For the family of intervals £int and all e > 0, [ipu+{tp~1 logT) — 1/2 - e\ < Aj(l) < •0u+(t/F1 logT) + 3/2 + £ for all sufficiently large T with probability 1. The proof is by direct calculation of good estimates of the relevant prob­abilities and application of standard Borel-Cantelli techniques. 1 Notice that A^(F, L) and Arj,(V, L) are not necessarily measurable as functions from the underlying probability space Q to R. This problem is circumvented by using outer probability, denoted as P*, for inequalities with these quantities. In particular, if Aj is a sequence of (not necessarily measurable) subsets of Q, we write lim PRAp) = 1 iff T—►oo lim P*(Q \ Aj) = 0, and if A C ÍÍ, we say that A a.s. (almost surely, or with probability T—► oo one) if there exists Qq C A with P(Qo) = 1-

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