We show that certain billiard tables with non-compact cusps are mixing with respect to the invariant infinite measure, in the sense of Krengel and Sucheston. Moreover, we show that the scaling rate is slower than a certain polynomial rate.

Applications of the original prophet inequalities of Krengel and Sucheston are made to problems of order selection, non-measurable stop rules, look-ahead stop rUles, and iterated maps of random variables. Also, proofs are given of two results of Hill and Hordijk c?ncerning optimal orderings of uniform and exponential d~stributions. §l. INTRODUCTION Universal inequalities comparing the two func­

has density one in Z with respect to some sequence of intervals Ik = [ak, bk] with bk−ak → ∞. (This means that d{Ik}(S) = lim k→∞ |S∩Ik| bk−ak+1 = 1.) A vector f ∈ H is called weakly wandering if there is an infinite set S ⊆ Z such that for any n,m ∈ S, n 6= m, one has 〈Uf, Uf〉 = 0. The following theorem due to U. Krengel gives a characterization of weak mixing in terms of weakly wandering vect...

The main purpose of this paper is to prove the following theorem, which sharpens results of Krengel and Sucheston [11, 12] in which the weaker constant 2(1 +-(3) was obtained. (Here EX is the expected value of the ran­ dom variable X, and ~ and T are the sets of stop rules ~ n, and of a.s. finite stop rules, respectively.) Theorem 1.1. If Xl' ... , X n are independent non-negative random variab...

We study rearrangements (Y1, . . . , Yn) = (Xσ1 , . . . ,Xσn) (where σ is a random permutation) of an i.i.d. sequence of random variables (X1, . . . ,Xn) uniformly distributed on [0, 1]; in particular we consider rearrangements satisfying the strong rank independence condition, that the rank of Yk among Y1, . . . , Yk is independent of the values of Y1, . . . , Yk−1. Nontrivial examples of such...

S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel,...