André Robert Dabrowski

An invariance principle for weakly associated random vectors

The positive dependence notion of association for collections of random variables is generalized to that of weak association for collections of vector valued random elements in such a way as to allow negative dependencies in individual random elements. An invariance principle is stated and proven for a stationary, weakly associated sequence of d-valued or separable Hilbert space valued random elements which satisfy a covariance summability condition.

FMRFamide and membrane stretch as activators of the Aplysia S-channel

The long-standing distinction between channels and transporters is becoming blurred, with one pump protein even able to convert reversibly to a channel in response to osmotic shock. In this light, it is plausible that stretch channels, membrane proteins whose physiological roles have been elusive, may be transporters exhibiting channel-like properties in response to mechanical stress. We recently described a case, however, where this seems an unlikely explanation. An Aplysia K channel whose physiological pedigree is well established (it is an excitability-modulating conductance mechanism) was found able to be activated by stretch. Here we establish more firmly the identity of this Aplysia conductance, the S-channel, as a stretch channel. We show that the permeation and fast kinetic properties of the stretch-activated channel and of the FMRFamide-activated S-channel are indistinguishable. We have also made progress in extending the kinetic analysis of the stretch channel to situations of multiple channel activity. This analysis implements a novel renewal theory approach and is therefore explained in some detail.

A functional law of the iterated logarithm for associated sequences

Recently, a functional central limit theorem and a Berry-Essen Theorem have been demonstrated for classes or associated random variables. Using these results, and similar results for multiplicative sequences, we show a functional law of the iterated logarithm for associated sequences satisfying a rate requirement.

Poisson limits for U-statistics

We study Poisson limits for U-statistics with non-negative kernels. The limit theory is derived from the Poisson convergence of suitable point processes of U-statistics structure. We apply these results to derive infinite variance stable limits for U-statistics with a regularly varying kernel and to determine the index of regular variation of the left tail of the kernel. The latter is known as correlation dimension. We use the point process convergence to study the asymptotic behavior of some standard estimators of this dimension.

A note on a theorem of Berkes and Philipp for dependent sequences

We improve an almost sure invariance principle for f-mixing sequences of real random variables with finite (2 + [delta])th moment (0 0.

An almost sure invariance principle for triangular arrays of banach space valued random variables

SummaryWe give a simpler proof of the probability invariance principle for triangular arrays of independent identically distributed random variables with values in a separable Banach space, recently proved by de Acosta [1], and improve this result to an almost sure invariance principle.

Return and Value at Risk using the Dirichlet Process

There exists a wide variety of models for return, and the chosen model determines the tool required to calculate the value at risk (VaR). This paper introduces an alternative methodology to model‐based simulation by using a Monte Carlo simulation of the Dirichlet process. The model is constructed in a Bayesian framework, using properties initially described by Ferguson. A notable advantage of this model is that, on average, the random draws are sampled from a mixed distribution that consists of a prior guess by an expert and the empirical process based on a random sample of historical asset returns. The method is relatively automatic and similar to machine learning tools, e.g. the estimate is updated as new data arrive.

Extremal point processes and intermediate quantile functions

SummaryWe prove an invariance principle in probability for planar point processes associated with extremal processes. The underlying sequence of random variables is absolutely regular, and satisfies a local asymptotic independence condition. A strong approximation for triangular arrays of such point processes is also stated. We apply these results to the weak convergence of intermediate quantile functions.

Jump diffusion approximation for a Markovian transport model

Publisher Summary This chapter analyzes a Markovian model for particle transport in a fluidized bed chemical reactor and proves a diffusion approximation. The transport model is basically a birth-death Markov process with reflection at the origin and occasional jumps to the origin, modeling transport in the wakes of rising fluidization bubbles. The chapter establishes here a strong approximation by a jump diffusion process with trajectory-dependent jump times. The chapter emphasizes on a diffusion process model for the motion of a particle in a fluidized bed reactor. A fluidized bed is obtained by forcing gas through the lower distributor plate—a plate permeable to the gas but not the powder. The transport model is basically a birth-death Markov process with reflection at the origin and occasional jumps to the origin, modelling transport in the wakes of rising fluidization bubbles. The chapter establishes here a strong approximation by a jump diffusion process with trajectory-dependent jump times.

Estimating conditional occupation-time distributions for dependent sequences

Consider a random integer-valued process X(t) on Z(+) that satisfies some weak dependence condition. We study the empirical distribution function of the occupation times of such a process and prove convergence to a suitable Gaussian process. An application to the statistical analysis of open and closed sojourn-time distributions for ion channels is provided.