The present paper is devoted to further development of commutative nonstationary field themes; the first studies in this area were performed by K. Kirchev and V. Zolotarev [4, 5]. In this paper a more complicated variant of commutative field with nonstationary rank 2, carrying into more general situation for correlation function is studied. A condition of consistency (see (7) below) for commutative field is placed in the basis of the method proposed in [4, 5] and developed in this paper. The following...

* Partially supported by Grant MM-428/94 of MESC.In this paper we present some generalizations of results of M. S. Livšic [4,6], concerning regular colligations (A1, A2, H, Φ, E, σ1, σ2, γ, ˜γ) (σ1 > 0) of a pair of commuting nonselfadjoint operators A1, A2 with finite dimensional imaginary parts, their complete characteristic functions and a class Ω(σ1, σ2) of operator-functions W(x1, x2, z): E → E in the case of an inner function W(1, 0, z) of the class Ω(σ1). ...

The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ${\int }_0^{2\pi }{\mathrm{e}}^{ix}\mathrm{d}\mu \left(x\right)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS – for example, the Minkowski sum – have natural translations in terms of probability measure operations,...

Let $(X_n,n\ge 1),({\tilde{X}}_n,n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${ℝ}^k$ and $S_n=X_1+\cdots +X_n$, ${\tilde{S}}_n={\tilde{X}}_1+\cdots +{\tilde{X}}_n$, $n\ge 1$. Assuming that $E{X}_1=E{\tilde{X}}_1$, $E|X_1{|}^2<\infty$, $E|{\tilde{X}}_1{|}^{k+2}<\infty$ and the existence of a density of ${\tilde{X}}_1$ satisfying the certain conditions we prove the following inequalities: $$v(S_n,{\tilde{S}}_n)\le cmax\{v(X_1,{\tilde{X}}_1),{\zeta }_2(X_1,{\tilde{X}}_1)\},n=1,2,\cdots ,$$ where $v$ and ${\zeta }_2$ are the total variation and Zolotarev’s metrics, respectively.

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist.36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab.28...

We consider the problem of optimal investment for maximal expected utility in an incomplete market with trading strategies subject to closed constraints. Under the assumption that the underlying utility function has constant sign, we employ the comparison principle for BSDEs to construct a family of supermartingales leading to a necessary and sufficient condition for optimality. As a consequence, the value function is characterized as the initial value of a BSDE with Lipschitz growth.