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...

The following corrections are required in Theorem 4.7.

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...

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...

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...

Following the model introduced by Aguech et al. (Probab Eng Inf Sci 21:133–141, 2007), the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyse weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...