Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

Oscillating processes with independent increments and nondegenerate Wiener component

For an oscillating process _z(t) (_z(0)=2,t0), which is defined with the help of two homogeneous processes ₁(t) and ₂(t) with independent increments and nondegemerate Wiener components, under certain restrictions we establish a relation of the form and find the characteristic function of the ergodic distribution of the process considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1415–1421, October, 1990.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Geometric modules and Quinn homology theory

LetR be a ring with unit and invariant basis property. In [1], the authors define a functorK(_;R):TOP/LIP _c-LPEP by combining the open cone construction of [7] with a geometric module construction and show this functor is a homology theory. This paper shows that if attention is restricted to objects TOP/LIP _c with a homotopy colimit structure, then the functorK(_;R) is a Quinn homology theory, In particular, for each having a homotopy colimit structure,K(;R) is a homotopy colimit in the category of -spectra. Furthermore, the constituent spectra of this homotopy colimit are obtained naturally from the fibres of .Partially supported by the National Science Foundation under grant number DMS88-03148.Partially supported by the SNF (Denmark) under grant number 11-7792.     

Limit distribution function of inhomogeneities in regions with random boundary. I

A study is made of the interaction of systems of charged particles with a membrane consisting of inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. A system of equations for the self-consistent potentialU ₁(x, ⁰,..., ^N ,...) and density of surface changes (x, ⁰,..., ^N ,...) is derived and solved.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 98–112, July, 1992.     

On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions

We fix a rich probability space (,F,P). Let (H,) be a separable Hilbert space and let be the canonical cylindrical Gaussian measure on H. Given any abstract Wiener space (H,B,) over H, and for every Hilbert–Schmidt operator T: HBH which is (|{}|,)-continuous, where |{}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process : (,F,P)×[0,1](B,|{}|) as a pathwise solution of the following infinite-dimensional Langevin equation d _t =db _t +T( _t )dt with the initial data ₀=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,). The richness of the probability space (,F,P) then implies the following consequences: the probability space is independent of the abstract Wiener space (H,B,) (in the sense that (,F,P) does not depend on the choice of the Gross-measurable norm |{}|) and the space C _B consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of . Finally, we present a way to obtain pathwise continuous solutions :d _t =

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Sequential estimation of the parameters of diffusion processes

For the parameter of a diffusion process(t), satisfying the stochastic differential equation d(t)=f (t,)dt+dw(l), we propose an effective sequential estimation plan with an unbiased and normally distributed estimate. The proposed sequential plan is discussed in detail for the example of a process (t) having a linear stochastic differential.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 627–638, November, 1972.In conclusion the author wishes to express his deep gratitude to A. N. Shiryaev for formulating the problem and for useful observations     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

The limit of the spectral radius of block Toeplitz matrices with nonnegative entries

A bi-infinite sequence ...,t _–2,t _–1,t ₀,t ₁,t ₂,... of nonnegativep×p matrices defines a sequence of block Toeplitz matricesT _n =(t _ik ),n=1,2,...,, wheret _ik =t _k–i ,i,k=1,...,n. Under certain irreducibility assumptions, we show that the limit of the spectral radius ofT _n , asn tends to infinity, is given by inf{()[0,]}, where () is the spectral radius of _jz t _j ^j .Supported by SFB 343 Diskrete Strukturen in der Mathematik, Universität Bielefeld     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Explicit expression for the transition density of the solution of a stochastic diffusion equation with piecewise-constant drift coefficient

The explicit form of the transition density is determined for the solution (t) of the stochastic diffusion equation d(t)=a((t))dt+dw(t), where a(z)= for x [a, b] and a(x)=0 for x [a, b], w(t) is a Wiener process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 99–105, 1987.     

On minimax filtration of vector processes

We study the problem of optimal linear estimation of the transformation of a stationary random process (t) with values in a Hilbert space by observations of the process (t) + (t) fort0. We obtain relations for computing the error and the spectral characteristic of the optimal linear estimate of the transformationA for given spectral densities of the processes (t) and (t). The minimax spectral characteristics and the least favorable spectral densities are obtained for various classes of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 389–397, March, 1993.     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

Comparison results for the lower tail of Gaussian seminorms

Let =( _n ) be i.i.d.N(0, 1) random variables andq(x), q(x):R [0, ) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q()<) andP(q()<) goes to a positive constant as 0⁺. We give satisfactory answers forl ₂-norms and also some results for sup-norms andl _p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL ₂-norms.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.