Volume 45, Issue 2

We prove Orlicz-space versions of Hardy and Landau–Kolmogorov inequalities for Gaussian measures on ℝ n .

A new measure of dependence called pseudo-covariance and related to covariance is proposed. It can be applied in problems when the classic covariance fails. We show that it can be used as a measure of dependence of uncorrelated random variables and in characterizations of continuous distributions (here power distribution on (0, 1) and exponential distribution).

We present a new elementary and elegant probabilistic proof of conditional version of the Donati-Martin and Yor formula, more precisely, the conditional version of the Laplace transform of ∫0tBu2du$\int_0^t {B_u^2 du} $ given B t = y , where B is a Brownian motion. Next, using our form of conditional formula, we obtain the new results concerning Laplace transforms of some processes and a kind of reflection principle.

Convergence in distribution, convergece in probability, and convergence almost surely, uniform with respect to a family of probability distributions , is considered. These concepts appeared to be appropriate tools for asymptotic theory of mathematical statistics and many partial results are scattered in the literature of the subject. The aim of this note is to present a unified review of the results in a general and abstract setup. We examine a few rather paradoxical examples which hopefully shed some light on the subtleties of the underlying definitions and the role of asymptotic approximations in statistics. A motivation for considering these problems is provided by their applications.

We investigate integrability properties of processes with linear regressions and quadratic conditional variances. We establish the right order of dependence of which moments are finite on the parameter στ defined below, raising the question of determining the optimal constant.

In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

The Nevalinna transform K a , ρ ( z ) of a positive measure ρ and a constant a , plays an important role in complex analysis and – more recently – in the context of the boolean convolution. We show here that its restriction to the imaginary axis, k a , ρ ( it ), can be expressed as the Laplace transform of the Fourier transform (a characteristic function) of ρ . Consequently, k a ρ is sufficient for the unique identification of the measure ρ and the constant a . Finally, we identify a relation between the free additive Voiculescu ⊞ and boolean ⊎ convolutions.

In the paper we use penalty method to approximate a number of general stopping problems over finite horizon. We consider optimal stopping of discrete time or right continuous stochastic processes, and show that suitable version of Snell’s envelope can by approximated by solutions to penalty equations. Then we study optimal stopping problem for Markov processes on a general Polish space, and again show that the optimal stopping value function can be approximated by a solution to a Markov version of the penalty equation.

Principle of Conditioning is a well-known heuristic rule which allows constructing limit theorems for sums of dependent random variables from existing limit theorems for independent summands. In the paper we state a general limit theorem on converegnce to stable laws, which is valid for stationary sequences and provides a link between the Principle of Conditioning and ergodic theorems.

Let X(μ)={Xt(μ);t≥0}$X^{(\mu )} = \{ X_t^{(\mu )} ;t \ge 0\} $, μ > 0, be the n -dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space ℍ n having the Laplace–Beltrami operator with drift as its generator. We prove the reflection principle for X ( μ ) which enables us to study the process X ( μ ) killed when exiting the hyperbolic half-space, that is the set D = { x ∈ ℍ n : x 1 > 0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X ( μ ) . Finally, we derive formula for the λ-Poisson kernel of the set D .

We investigate conditional stable processes in a Lipschitz domain D and conditional stable processes in the image of D under the Kelvin transform. We show that, with a suitable change of time, these processes are equal in distribution. As an application, we show the equivalence of the Hardy spaces and the relative Fatou theorem for D and its image.

In the present note we discuss the conditions under which for a sequence of nonnegative i.i.d. random variables ( X n ) n ∈ℕ with infinite mean, there exist sequences of nonnegative real numbers ( g ( n )) n ∈ℕ and ( h ( n )) n ∈ℕ such that lim⁡n→∞1g(n)∑k=1nXkh(k)=1 a.s.$$\mathop {\lim }\limits_{n \to \infty } {1 \over {g(n)}}\sum\limits_{k = 1}^n {{{X_k } \over {h(k)}} = 1\;a.s.} $$

In this note we show that for Z being a birth and death process on ℤ or Brownian motion with drift and ℰ′ = (0, ∞), the speed of convergence to the quasistationary distribution is of order 1 /t . The corresponding version that X is the number of calls in M/M/1 queue or the reflected Brownian motion is also considered. The result is obtained by asymptotic expansions of some transition functions. For this we use some new asymptotic expansion of the Bessel function.

The main purpose of this paper is to extend our previous construction of T -Fock spaces from a given Yang–Baxter operator satisfying the inequalities −1 ≤ T ≤ 1 to the constructions of T -symmetric Fock spaces related to the class of Yang–Baxter–Hecke operators meeting a weaker condition that T ≥ −1. The new representation of the monotone Fock space of N. Muraki will be given. The main idea of this paper is the new class of generalized Gaussian random variables acting on suitable T -symmetric Fock spaces. Relations with the row and column operator space will be also given.

A single period, zero-sum, multi-player game is constructed. Each player can either exit the game for a fixed payoff or stay and split the remaining payoff with the other non-exiting players. The emphasis is put on the rivalrous nature of the payoffs, meaning that the sum of all payoffs is fixed, but the exact allocation is based on the players’ decisions. The value at which Nash and optimal equilibria are attained is shown to be unique and it is constructed explicitly.

In the paper we solve a problem of optimal stopping of a risk process in two alternative settings. We assume that the main characteristics of the risk process change according to unobservable random variable. In the first model we assume that the post-disorder distributions are not known a’priori and are randomly chosen from a finite set of admissible distributions. The second model concentrates on a situation when more than one disorder is possible. For both models optimal stopping rules with respect to given utility function are constructed using dynamic programming methodology.

We prove that in a non-atomic probability space, for a sequence of any integrable r.v. ( X n ) we have the following equivalence: 𝔼( X n |𝒰) → 0 a.s. for any σ-field 𝒰 of events iff X n → 0 a.s. and 𝔼 sup n ≥1 | X n | < ∞.

Some recent results concerning generation and asymptotic properties of stochastic semigroups are presented. The general results are applied to biological models described by piecewise deterministic Markov processes: birth-death processes, the evolution of the genome, genes expression and physiologically structured models.