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

Studio Scientiarum Mathematicarum Hungarica 31 (1996), 1-13 LIMIT LAWS FOR THE MAXIMAL AND MINIMAL INCREMENTS OF THE POISSON PROCESS P. AUER and K. HORNIK Dedicated to Professor P. Révész for his 60th birthday Abstract We give a partial survey of strong laws for maximal and minimal increments of homo­geneous Poisson processes in R^, with emphasis on the cases where exact rates have been obtained. We also cite closely related results for the increments of the empirical process. Very recent results on the L-increments of a Poisson process are given, where an L-incre­­ment is the sum of points in L disjoint sets. Finally, we point out possible generalizations of the results and directions of future research. 1. Introduction and notations Suppose we are given a random collection of points in that follow a homogeneous Poisson process on Rd with parameter ifi, i.e., if for every Borel subset A of R^, 77(A) denotes the number of points contained in A and A(A.) its Lebesgue measure, then 77(A) has a Poisson distribution with parameter ■0A(A), and the numbers of points in disjoint sets are jointly independent. In this paper, we shall survey results on the asymptotic behaviour as T —> —f oo of the maximal and minimal number of points in certain families of sets of volume Vj- contained in Hx — [0, T]d, where 0 < Vx ^ Td and lim Vx/Td = T —KX> = 0. Let Í be a family of Borel measurable subsets of [0, l]d, and let the maximal and minimal numbers be defined as A+(V) = max{7?(£) :E£T<£, A(E) = V} Aj{V) = mm{r](E) : Ee T<£,\{E) = V). (If t £ R, and E Q Rd, then tE = {tx : x £ E}; similarly, t<£ = {tE: E £ £}.) To investigate the behaviour of X-increments we also define Aj(F, X) = m&x{ri(Ei U • • ■ U Ei) : E{ £ T£, A(£,) = V, {E,} disjoint }, Aj(P, X) = min{r/(Xi U • • • U El) : Ei £ T£, A(E{) = V, {Et} disjoint }, 0081-6906/96/$ 5.00 ©1996 Akadémiai Kiadó, Budapest MAGYAR tudományos AKADÉMIA - KÖNYVTÁRA 1991 Mathematics Subject Classification. Primary 60G55; Secondary 60F15. Key words and phrases. Homogeneous d-dimensional Poisson process, empirical pro­cess, maximal and minimal increments, L-increments, strong laws, asymptotically quasi­­deterministic.

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