Journal of Theoretical Probability

http://link.springer.com/journal/10959

List of Papers (Total 52)

Fractal-Dimensional Properties of Subordinators

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely \(\lim _{\delta \rightarrow 0}U(\delta )N(t,\delta ) = t\), where \(N(t,\delta )\) is the minimal number of boxes of size at most \( \delta \) needed to cover a subordinator’s range ...

Correction to: An Itō Formula in the Space of Tempered Distributions

The following corrections are required in Theorem 4.7.

From Low- to High-Dimensional Moments Without Magic

We aim to compute the first few moments of a high-dimensional random vector from the first few moments of a number of its low-dimensional projections. To this end, we identify algebraic conditions on the set of low-dimensional projectors that yield explicit reconstruction formulas. We also provide a computational framework, with which suitable projectors can be derived by solving ...

A Behavioral Interpretation of Belief Functions

Shafer’s belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer’s belief functions via a betting ...

Two Applications of Random Spanning Forests

We use random spanning forests to find, for any Markov process on a finite set of size n and any positive integer \(m \le n\), a probability law on the subsets of size m such that the mean hitting time of a random target that is drawn from this law does not depend on the starting point of the process. We use the same random forests to give probabilistic insights into the proof of ...

Segregating Markov Chains

Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There are, however, discrete finite (reducible) Markov chains, for which two copies started in different states can be coupled to meet almost surely in finite time, yet their ...

Heavy-Tailed Random Walks on Complexes of Half-Lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution \(\mu _k\). If \(\chi _k\) is 1 for one-sided ...

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system ...

Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$where \(\tau _{\alpha }\), \(\alpha \in [m]\), are real numbers and ...

Large Deviations of Continuous Regular Conditional Probabilities

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for ...

Turning a Coin over Instead of Tossing It

Given a sequence of numbers \((p_n)_{n\ge 2}\) in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability \(p_n\), \(n\ge 2\), independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as \(n\rightarrow \infty \)? We show ...

Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a ...

Power Law Condition for Stability of Poisson Hail

The Poisson hail model is a space-time stochastic system introduced by Baccelli and Foss (J Appl Prob 48A:343–366, 2011) whose stability condition is nonobvious owing to the fact that it is spatially infinite. Hailstones arrive at random points of time and are placed in random positions of space. Upon arrival, if not prevented by previously accumulated stones, a stone starts ...

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) ...

Spatial Central Limit Theorem for Supercritical Superprocesses

We consider a measure-valued diffusion (i.e., a superprocess). It is determined by a couple \((L,\psi )\), where L is the infinitesimal generator of a strongly recurrent diffusion in \(\mathbb {R}^{d}\) and \(\psi \) is a branching mechanism assumed to be supercritical. Such processes are known, see for example, (Englander and Winter in Ann Inst Henri Poincaré 42(2):171–185, 2006), ...

On Some Properties of a Class of Fractional Stochastic Heat Equations

We consider nonlinear parabolic stochastic equations of the form \(\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }\) on the ball \(B(0,\,R)\), where \(\dot{\xi }\) denotes some Gaussian noise and \(\sigma \) is Lipschitz continuous. Here \(\mathcal {L}\) corresponds to a symmetric \(\alpha \)-stable process killed upon exiting B(0, R). We will consider two types of ...

An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes

In a recent paper in this journal, Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor’s classical results on first-hitting times of a birth–death process on the nonnegative integers by establishing a representation for the Laplace transform \({\mathbb {E}}[e^{sT_{ij}}]\) of the first-hitting time \(T_{ij}\) for any pair of states i and j, as well ...

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

The Matsumoto–Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Wesołowski (Ann Probab 28:1371–1383, 2000), the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we ...

Banach Random Walk in the Unit Ball \(S\subset l^{2}\) and Chaotic Decomposition of \(l^{2}\left( S,{{\mathbb {P}}}\right) \)

A Banach random walk in the unit ball S in \(l^{2}\) is defined, and we show that the integral introduced by Banach (Theory of the integral. Warszawa-Lwów, 1937) can be expressed as the expectation with respect to the measure \({{\mathbb {P}}}\) induced by this walk. A decomposition \(l^{2}\left( S,{{\mathbb {P}}}\right) =\bigoplus _{i=0}^{\infty } {{\mathfrak {B}}}_{i}\) in terms ...

Uniquely Determined Uniform Probability on the Natural Numbers

In this paper, we address the problem of constructing a uniform probability measure on \({\mathbb {N}}\). Of course, this is not possible within the bounds of the Kolmogorov axioms, and we have to violate at least one axiom. We define a probability measure as a finitely additive measure assigning probability 1 to the whole space, on a domain which is closed under complements and ...

The Lukacs–Olkin–Rubin Theorem on Symmetric Cones Without Invariance of the “Quotient”

We prove the Lukacs–Olkin–Rubin theorem without invariance of the distribution of the “quotient,” which was the key assumption in the original proof of (Olkin–Rubin in Ann Math Stat 33:1272–1280, 1962). Instead, we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) “quotient” for any division algorithm ...