Stochastic processes with independent increments for random mappings

Using additive functions defined on the combinatorial structure of all mappings of anN set into itself, we define paths in the space endowed with the Skorokhod topology. Taking a mapping with equal probability, we get a sequence of random processes. Necessary and sufficient conditions for the weak convergence of this sequence to a stochastic process with independent increments are established. It is shown that the class of such processes contains all possible limits, provided that, on the components of a mapping, the additive functions have values small in average. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 498–516, October–December, 1999. Translated by E. Manstavičius     

On the intrinsic normalizations of the seminonholonomic complexesSN Gr(m, n, (m+1)(n−m)−ϱ). I

Some intrinsic normalizations of seminonholonomic complexes of then-dimensional projective space are considered. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 517–538, October–December, 1999. Translated by R. Lapinskas     

On one-dimensional stochastic differential equations driven by stable processes

We consider the one-dimensional stochastic differential equation dX _t=b(t, X_t−) dZ _t, whereZ is a symmetric α-stable Lévy process with α ε (1, 2] andb is a Borel function. We give sufficient conditions under which the equation has a weak nonexploding solution. Partially supported by Programma Professori Visitatori of G. N. A. F. A. (Italy). Partially supported by MURST (Italy). The present research was completed while the second author was visiting the Institute of Mathematics and Informatics (Vilnius, Lithuania) in spring of 1999. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 361–385, July–September, 2000. Translated by H. Pragarauskas     

Intrinsic normalizations of a hyperplane distribution on the Grassmann manifold. I

We consider intrinsic normalizations of a hyperplane distribution on the Grassmann manifold of ann-dimensional projective space. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 257–273, April–June, 1999. Translated by V. Mackevičius     

Infinitary calculus for a restricted first-order linear temporal logic without contraction on quantified formulas

An infinitary calculus for a restricted fragment of the first-order linear temporal logic is considered. We prove that for this fragment one can construct the infinitary calculusG _Lω ^* without contraction on predicate formulas. The calculusG _Lω ^* possesses the following properties: (1) the succedent rule for the existential quantifier is included into the corresponding axiom; (2) the premise of the antecedent rule for the universal quantifier does not contain a duplicate of the main formula. The soundness and completness ofG _Lω ^* are also proved. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 378–397, July–September, 1999. Translated by R. Lapinskas     

The von mises theorem for farey fractions

We consider systems of strongly additive functions defined on the set of rational numbers and taking the values 0 and 1 on prime elements of ℚ. For these systems, we obtain necessary and sufficient conditions for the weak convergence to the Poisson law. Research supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 425–439, July–September, 1999. Translated by J. <Siaulys     

On weak convergence of random fields

We provide a characterization of compactness in the spaceD of functions of two variables defined on a unit square. The functions fromD have the property that their discontinuity points lie on smooth curves. Conditions for the tightness of probability measures inD and conditions for weak convergence of random fields with trajectories inD are derived. Vilnius Gediminas Technical University, Saulétekio 11; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp 169–184, April–June, 1999. Translated by R. Banys     

A sieve result for Farey fractions

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ₁, λ₂). An asymptotic formula for the sifting function is derived under the assumption that (λ₁, λ₂)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p ₁ p ₂...p _s ,p ₁≥p ₂>-..., ξk)_is defined by ξ(k)=(logp ₁/logk, logp ₂/logk,..., logp _s /logk, 0, ...); allp _i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108–127, January–March, 1999. Translated by V. Stakénas     

On convergence to the poisson law for distributions of the sum of random variables connected into a chain

This paper deals with the triangular array of random variables where every sequence is connected into a stationary Markov chain. For the uniformly strongly mixing chain we find conditions under which the distributions of the sum converge to the Poisson law. A more general case of theL ^p-regular Markov chain is also considered. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 297–307, July–September, 1995. Translated by R. Lapinskas     

Some Properties of the Functor O β

It is proved that the space O_β(X) of weakly additive order-preserving normed functionals with compact supports is a convex subset of the space C_p(C_b(X)). Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 313, 2004, pp. 131–134.     

Behavior of the homogeneous symmetric polynomial in an angular region

Vilnius Technical University, Saulétekio al. 11, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys Vol. 35, No. 4, pp. 469–488, October–December, 1995.     

Some estimates of thenth difference quotient of a function analytic in a convex region

Vilnius Technical University, Saulètekio 11, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 4, pp. 489–501, October–December, 1995.     

Large deviations for integer centered poisson approximation

Large deviations in a centered Poisson approximation are examined. For lattice distributions, the centered Poisson approximation is more universal than the normal and standard Poisson laws. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 9–23, January–March, 1999. Translated by V. Čekanavičius     

Some remarks on subfields of algebraic number fields

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 2 pp. 231–232, April–June, 1995     

An approximation of a nonlinear integral equation driven by a function of boundedp-variation

The uniform distance between the solution of a nonlinear equation driven by a functionh with boundedp-variation and its Milstein-type approximation is estimated by δ _n v γ _p (n) v γ _p ² (n), where δ _n =max(t _k −t _k−1 ) is the maximum step size of the approximation on the interval [0,T], γ _p (n)=max υ _p ^1/p (h;[t _k-1,t _k ]), 1 <p < 2, and υ _p (h;[t _k-1,t _k ]) is thep-variation of the functionh on [t _k-1,t _k]. In particular, ifh is a Lipschitz function of order α, then the uniform distance has the bound δ _n ^α for δ_n <1. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius Technical University, Saulétekio 11, 2054 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 317–330, July–September, 1999.     

On a problem of A. M. Vasil'ev

Vilnius Pedagogical University, Studentu 39, 2034 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 3, pp. 388–407, July–September, 1994.     

On the blow-up of solutions of a system of nonlinear Schrödinger equations

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 181–189, April–June, 1995.     

Difference scheme for the nonlinear problem of magnetization

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 32, No. 2, pp. 237–248, April–June, 1992.     

Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. II

Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 1, pp. 94–106, January–March, 1994.     

Locally minimax efficiency of nonparametric density estimators for χ2-type losses

In the paper, the asymptotic expression of the locally minimax risk with weightedL ₂ losses is found. It is shown that in the case of χ²-type losses the expression is independent of the center of contracting neighborhoods. A locally minimax estimator for the χ²-type losses is constructed. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 398–424, July–September, 1999.