Ronald Pyke

The weak convergence of the empirical process with random sample size

In many applied probability models, one is concerned with a sequence { X n : n > 1} of independent random variables (r.v.s) with a common distribution function (d.f.), F say. When making statistical inferences within such a model, one frequently must do so on the basis of observations X 1 , X 2 ,…, X N where the sample size N is a r.v. For example, N might be the number of observations that it was possible to take within a given period of time or within a fixed cost of experimentation. In cases such as these it is not uncommon for statisticians to use fixed-sample-size techniques, even though the random sample size, N , is not independent of the sample. It is therefore important to investigate the operating characteristics of these techniques under random sample sizes. Much work has been done since 1952 on this problem for techniques based on the sum, X 1 + … + X N (see, for example, the references in (3)). Also, for techniques based on max( X 1 , X 2 , …, X N ), results have been obtained independently by Barndorff-Nielsen(2) and Lamperti(9).

The existence of set-indexed Lévy processes

SummaryThis paper considers the problem of the existence of set-indexed Lévy processes having regular sample paths defined over as large a class, A, as possible of subsets of the unit cube in ℝd. Regular sample paths means here the natural generalization of right continuity and left limits, to concepts of outer continuity and inner limits. A general integral condition involving the Lévy measure and the entropy exp(H(δ)) of the class A is obtained that is sufficient for the existence of such regular processes. In the particular case where the process is stable of index α, α∈(1, 2), the condition becomes % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9qq-Jar% pepeea0xd9q8as0-LqLs-Jirpepeea0-as0Fb9pgea0-arFve9Fve9% Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdXbqaaiaacI% cacaWGibGaaiikaiaadIhacaGGPaGaai4laiaadIhacaGGPaWaaWba% aSqabeaacaaIXaGaeyOeI0IaaGymaiaac+cacqaHXoqyaaGccaWGKb% GaamiEaiabgYda8iabg6HiLcWcbaGaaGimaaqaaiaaigdaa0Gaey4k% Iipaaaa!49D3!

A central limit theorem for D(A)-valued processes

\int\limits_0^1 {(H(x)/x)^{1 - 1/\alpha } dx < \infty }

The Haar-Function Construction of Brownian Motion Indexed by Sets*

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Multidimensional Empirical Processes: Some Comments*

The Brownian motion with multi-dimensional time parameter introduced by Paul Levy can be viewed as a set-indexed Brownian process with independent increments. This is demonstrated in a way which yields a unified representation of Levys Brownian motion and the Brownian sheet. Levys Brownian motion, like Brownian sheet, is shown to be a special case of the additive set-indexed Gaussian process {Z(A): A [epsilon] A} with Cov(Z(A), Z(B) =[mu](A [intersection] B) for some measure [mu]. A particular family of spheres is seen to play the same basic role in this representation as the family of orthants plays for Brownian sheet. A related central limit theorem and invariance result are discussed for a natural family of empirical-like processes, indexed by large families of sets A.

Uniform pointwise ergodic theorems for classes of averaging sets and multiparameter subadditive processes

Let D(A) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D(A) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Levy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.

The Asymptotic Relative Efficiency of Goodness-Of-Fit Tests against Scalar Alternatives

Consider a two-lane road which is intersected on one side by a single-lane secondary road. A single car waiting on the secondary road may merge into either the nearest or the farthest lane. It is assumed that the traffic in each lane is independent of the other lanes and that the inter-arrival times of cars at the intersection in their respective lanes is exponential. The main purpose of this paper is to study the queue size on the secondary road when the secondary road has a special right-turn lane (or left-turn lane in some countries) which allows some cars to merge into the nearest lane of the main road even when other cars waiting to enter the far lane are present. The problem is approached by first setting up a four-state Markov Renewal process to describe the traffic on the main road. Next the merging process on the secondary road is described as a Markov Renewed process with a random environment. The event that the queue is empty is studied, and conditions are stated under which this event is recurrent or transient. Finally, the quantities which occur in the conditions for recurrence of an empty queue are derived explicitly for a one-car right turn lane.

The Limiting Distribution of the Rank Correlation Coefficient {R_g}

SummaryThe Haar-function construction of Brownian Motion is obtained when the process is indexed by families of ‘smooth’ subsets of the k-dimensional cube. As the uniform limit of signed-measures, the resultant process is a Gaussian process with continuous paths with respect to the Hausdorff metric. A class of related Gaussian set functions defined on these subsets is also introduced.