Necessary conditions for characterization of laws via mixed sums

SupposeX andY are independent and identically distributed, and independent ofU which satisfies 0U1. Recent work has centered on finding the lawsL(X) for whichX U(X+Y) where denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function ofX has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when –logU has a lattice law. A continuous version of the basic problem replaces sums with a Lévy process. Instead we consider self-similar processes, showing that a solution exists only whenU is constant, and then all processes of a given order are admitted.This research was in part supported by NSERC grant A-8466.     

The limiting distribution for the infinitely deep dam with a Markovian input

The paper outlines a case for taking greater interest in the bottomless, or infinitely deep, dam model in Hydrology. It then shows that for such a model with unit withdrawals and an ergodic Markov chain input process the limiting distribution of depletion, when this exists, is a zero modified geometric distribution. This result generalises the well known result for independent inputs. The technical conditions required for the proof are satisfied for finite state space input processes and are shown to be satisfied by certain infinite state space input processes. These include as special cases examples which have a negative binomial limiting input distribution.     

Limit theorems for record counts and times in the F α -scheme

We determine the almost sure and central limiting behaviour of the number of records and record times for the F ^α -scheme. Elementary methods are used to obtain general results. The basic results are extended to a random environments model.     

Limit theorems for numbers of near-records

Observations occurring between successive record times and within a distance a > 0 of the current record value are called near-records. Limit theorems for the number ξ _n (a) of near records are found for cases in which the parent distribution lies in a maximal domain of attraction and a is a function of n. Corollaries are indicated for numbers of near-k-records and sums of near-records. If the parent law is thin-tailed and a is constant, then a centered and normed version of logξ _n (a) has a limit law under appropriate conditions.      

Limit laws for the number of near maxima via the Poisson approximation

Given a sequence of i.i.d. random variables, new proofs are given for limit theorems for the number of observations near the maximum up to time n, as n → ∞. The proofs rely on a Poisson approximation to conditioned binomial laws, and they reveal the origin in the limit laws of mixing with respect to extreme value laws. For the case of attraction to the Fréchet law, the effects of relaxing a technical condition are examined. The results are set in the broader context of counting observations near upper order statistics. This involves little extra effort.     

A kinematical study of Kirchhoff-Rayleigh flow

Summary A method of calculating the separated flow of a viscous fluid is proposed, which allows to split up properly the boundary condition problem from the viscous phenomena. The theory is developed for the flow past a plate and yields wakes of finite extension having an underpressure which depends directly on the amount of vorticity diffusion and dissipation occurring in the fluid. Application of the method to real flows shows good agreement between the calculated and the measured velocity distributions in front of the plate and in the wake.

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Résumé Une méthode de calcul de l'écoulement décollé d'un fluide visqueux est proposée qui permet de séparer clairement le problème aux limites des phénomènes visqueux. La théorie est développée pour l'écoulement autour d'une plaque et donne des sillages de longueur finie ayant une dépression de culot directement dépendante de l'intensité de la diffusion et dissipation de la vorticité se produisant dans le fluide. L'application de la méthode à des écoulements réels montre une bonne concordance entre les répartitions de vitesse calculées et mesurées sur le devant de la plaque et dans le sillage.