Probability and Mathematical Statistics

Spis treści

2020-02-05T20:36:14+01:00

Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality

2020-03-09T13:57:07+01:00

This paper studies variance functions of Cauchy–Stieltjes Kernel CSK families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate CSK families with polynomial variance functions to generalized orthogonality.
Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.

Embedded Markov chain approximations in Skorokhod topologies

2020-03-09T13:57:04+01:00

We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner.
To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.
Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.

Karhunen–Loève decomposition of Gaussian measures on Banach spaces

2020-03-09T13:57:02+01:00

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.

On the exact dimension of Mandelbrot measure

2020-03-09T13:57:00+01:00

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of the argument used by Kahane 1987 on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J . As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets Eα of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

On the distribution and q-variation of the solution to the heat equation with fractional Laplacian

2020-03-09T13:56:57+01:00

We study the probability distribution of the solution to the linear stochastic heat equation with fractional Laplacian and white noise in time and white or correlated noise in space. As an application, we deduce the behavior of the q-variations of the solution in time and in space.

Wishart laws and variance function on homogeneous cones

2020-03-09T13:56:55+01:00

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

Admissible and minimax estimation of the parameters of the selected normal population in two-stage adaptive designs under reflected normal loss function

2020-03-09T13:56:53+01:00

In clinical research, one of the key problems is to estimate the effect of the best treatment among the given k treatments in two-stage adaptive design. Suppose the effects of two treatments have normal distributions with means θ1 and θ2, respectively, and common known variance σ2. In the first stage, random samples of size n1 with means X1 and X2 are chosen from the two populations. Then the population with the larger or smaller sample mean XM is selected, and a random sample of size n2 with mean YM is chosen from this population in the second stage of design. Our aim is to estimate the mean θM or θJ of the selected population based on XM and YM in two-stage adaptive design under the reflected normal loss function. We obtain minimax estimators of θM and θJ, and then provide some sufficient conditions for the inadmissibility of estimators of θM and θJ. Theoretical results are augmented with a simulation study as well as a real data application.

Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

2020-03-09T13:56:51+01:00

Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈ ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.

Stationarity as a path property

2020-03-09T13:56:49+01:00

Traditionally, stationarity refers to shift invariance of the distribution of a stochastic process. In this paper, we rediscover stationarity as a path property instead of a distributional property. More precisely, we characterize a set of paths, denoted by A, which corresponds to the notion of stationarity. On one hand, the set A is shown to be large enough, so that for any stationary process, almost all of its paths are in A. On the other hand, we prove that any path in A will behave in the optimal way under any stationarity test satisfying some mild conditions.

Estimates of the transition densities for the reflected Brownian motion on simple nested fractals

2020-03-09T13:56:47+01:00

We give sharp two-sided estimates for the functions gMt, x, y and gMt, x, y − gt, x, y, where gMt, x, y are the transition probability densities of the reflected Brownian motion on an Mcomplex of order M ∈ Z of an unbounded planar simple nested fractal and gt, x, y are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.

A two-parameter extension of Urbanik’s product convolution semigroup

2020-03-09T13:56:44+01:00

We prove that sna, b = Γan + b/Γb, n = 0, 1, . . ., is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers sna, bc, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin 2019 in the case b = 1. We describe a product convolution semigroup τca, b, c > 0, of probability measures on the positive half-line with densities eca, b and having the moments sna, bc. We determine the asymptotic behavior of eca, bt for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and Lopez 2015 and lead to a convolution semigroup of probability densities gca, bxc>0 on the real line. The special case gca, 1xc>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gca, bx lead to determinate Hamburger moment problems.

On the carrying dimension of occupation measures for self-affine random fields

2020-03-09T13:56:43+01:00

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle 1984, 1988, 1990, 1991. Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.