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

LIMIT LAWS Thus the maximal number of points contained in some interval of length 1 in [0, T] is one of the four values f(T) - 1, f(T), f(T) + 1, f(T) + 2, where f{T) = [ipu+(ip-1 log T)\ ~ (log T)/(log log T). In [14], Révész introduced the terms asymptotically quasi-deterministic and asymptotically deterministic. Definition 2.2. Let Yp be a random process and suppose that there exist deterministic functions ß\(T), /^(T) such that ßi(T)<Yp<ß2(T) for all sufficiently large T with probability 1. If /32(T) — ß\(T) = 0(1) as T —>oo, Yt is asymptotically quasi-deterministic (AQD)\ if ß2(T) — ß\ (T) = = o(l) as T —»oo, it is asymptotically deterministic (AD). Clearly, Theorem 2.1 implies that Aj(l) is AQD. 2.1. Generalization to the multi-dimensional case. A straightforward generalization of Theorem 2.1 was given by Auer, Hornik and Révész [3]. Theorem 2.3. Let Vp be eventually monotone and Vp — o(logT) for T —>oo. Then for the family of axis-parallel cubes and any e > 0, for all sufficiently large T with probability 1, where The proof follows the line of the proof of Theorem 2.1. Observe that Theorem 2.1 is a corollary of Theorem 2.3. Furthermore, note that Theorem 2.3 holds for quite general Vp below the Erdös-Rényi range2 (Vp log T). Putting some more restriction on Vp, we obtain the following 7 ={ Vpifu+ (\og {Td/VT)\ V Vr ) (1/2 + £•) log log T ^A+(Tt) + (7 + £) l°g((log Td)/Vr) log log T v vTif ) ' w ' “yiog((iogrd)/Fr) d+ 1/2 d + 3/2 if Vp is nondecreasing, if Vp is decreasing. 2 We refer to Deheuvels [8], Borovkov [6], and references therein for details concerning the original Erdös-Rényi [12] law and its extensions. The Erdös-Rényi range corresponds to when V = Vp is such that Vp/log T —* C with 0 < C < oo, and we will say that the increments are above (resp. below) the Erdös-Rényi range when C — oo (resp. C = 0).

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