This work concerns a class of discrete-time, zero-sum games with two players and Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I and, if the system is no halted, player I selects an action to drive the system and receives a running reward from player II. Measuring the performance of a pair of decision strategies by the total expected discounted reward, under standard continuity-compactness conditions it is shown...

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.

A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space $𝐃$ [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.

In recent joint papers with B. Schweizer, we used the notion of a copula to introduce a family of symmetric, nonparametric measures of dependence of two random variables. Here, we present n-dimensional extensions of these measures and of Spearman's ro. We study them vis-a-vis appropriate higher dimensional analogues of Rényi's axioms for measures of dependence, determine relations among them, and in some cases establish reduction formulae for their computation.

We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1+Θ(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the euclidean model.

Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables such that $E{X}_n=a_n$, $E{(X_n-a_n)}^2={\sigma }_n^2$, $n\ge 1$. Let $\{N_n,n\ge 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}={\sum }_{k=1}^{N_n}{X}_k$, $L_n={\sum }_{k=1}^n{a}_k$, $s_n^2={\sum }_{k=1}^n{\sigma }_k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,n\ge 1\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

Several new multivariate negative dependence concepts such as negative upper orthant dependent in sequence, negatively associated in sequence, right tail negatively decreasing in sequence and upper (lower) negatively decreasing in sequence through stochastic ordering are introduced. These concepts conform with the basic idea that if a set of random variables is split into two sets, then one is increasing whenever the other is decreasing. Our concepts are easily verifiable and enjoy many closure...

We provide explicit information geometric tubular neighbourhoods containing all bivariate distributions sufficiently close to the cases of independent Poisson or Gaussian processes. This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions and of the 5-manifold of bivariate Gaussians. We provide also the α-geometry for both manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases for a wide range of bivariate distributions; the...

Atsuji proposed some integrals along Brownian paths to study the Nevanlinna characteristic function T(f,r) when f is meromorphic in the unit disk D. We show that his criterios does not apply to the basic case when f is a modular elliptic function. The divergence of similar integrals computed along the geodesic flow is also proved. (A)