How many samples does it take to see all the balls in an urn?

Let an urn containN balls, numbered from 1 toN. A random number of balls are drawn without replacements from the urn, their numbers are noted and the balls are then returned to the urn. This is done repeatedly, the sample sizes being independent identically distributed. Letv be the number of samples needed to see all the balls. A simple approximation forEv and the asymptotic distribution ofv asN are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 58–63, July, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00387.     

Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X _{n, m}; (n, m) N ²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N ² is studied. There is a gap between the moment conditions for CLIL(N ¹) and those for CLIL(N ²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N ^r (, ) = {(n, m) N ²; n m n exp{(log n) ^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.     

Random Walks on Trees and an Inequality of Means

We define trees generated by bi-infinite sequences, calculate their walk-invariant distribution and the speed of a biased random walk. We compare a simple random walk on a tree generated by a bi-infinite sequence with a simple random walk on an augmented Galton-Watson tree. We find that comparable simple random walks require the augmented Galton-Watson tree to be larger than the corresponding tree generated by a bi-infinite sequence. This is due to an inequality for random variables with values in [1, [ involving harmonic, geometric and arithmetic mean.     

Metastates in Disordered Mean-Field Models II: The Superstates

We continue to investigate the size dependence of disordered mean-field models with finite local spin space in more detail, illustrating the concept of superstates as recently proposed by Bovier and Gayrard. We discuss various notions of convergence for the behavior of the paths (t_[tN]()) _{t(0, 1]} in the thermodynamic limit N. Here _n () is the Gibbs measure in the finite volume {1,..., n} and is the disorder variable. In particular we prove refined convergence statements in our concrete examples, the Hopfield model with finitely many patterns (having continuous paths) and the Curie–Weiss random-field Ising model (having singular paths).     

Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X _R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X _R is a random variable, depending on the environment. In dimension d = 1, the variable X _R converges in distribution to the Bernoulli variable, X = 0, 1 with equal probability, as R . Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.     

Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation

A quasi-isochronous vibroimpact system is considered, i.e. a linear system with a rigid one-sided barrier, which is slightly offset from the system's static equilibrium position. The system is excited by a sinusoidal force with disorder, or random phase modulation. The mean excitation frequency corresponds to a simple or subharmonic resonance, i.e. the value of its ratio to the natural frequency of the system without a barrier is close to some even integer. Influence of white-noise fluctuations of the instantaneous excitation frequency around its mean on the response is studied in this paper. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the application of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged stochastic equations are solved exactly by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset and small random variations of amplitude. Therefore, the analytical results may be expected to be adequate for small values of excitation/system bandwidth ratio or for small intensities of the excitation frequency variations. However, at very large values of the parameter the results are approaching those predicted by a stochastic averaging method. Moreover, Monte-Carlo simulation has shown the moment closure results to be sufficiently accurate in general for any arbitrary bandwidth ratio. The basic conclusion, both of analytical and numerical simulation studies, is a sort of smearing of the amplitude frequency response curves owing to disorder, or random phase modulation: peak amplitudes may be strongly reduced, whereas somewhat increased response may be expected at large detunings, where response amplitudes to perfectly periodic excitation are relatively small.     

RUC-Bases in Orlicz and Lorentz Operator Spaces

If is an RUC-basis in somecouple of non-commutative L _p-spaces, then is an RUC-basic sequence in any non-commutative Orlicz or Lorentz space which is an interpolation space for this couple.     

Projection Method for Moment Bounds on Order Statistics from Restricted Families: II. Independent Case

The method of projection, proposed in Part I, is applied to derive sharp moment bounds for the expectations of order statistics based onindependentsamples from restricted families of distributions. Three families are considered: life distributions with decreasing failure density, decreasing failure rate, and symmetric unimodal ones. The respective bounds are also numerically compared with those for general populations in both the dependent and independent cases.     

Weak laws of large numbers for cooperative gamblers

Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation. Work was supported in part by the Hungarian Scientific Research Fund, Grant T048360, and carried out within the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.