One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their application to the law of the iterated logarithm in Banach lattices is given. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 653–665, May, 2006.     

A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.      

Distribution of the supremum of random processes from quasi-BanachK σ-spaces

We study random processes from quasi-BanachK _σ-spaces of random variables whose domain of definition is not necessarily a compact set. We establish conditions under which the supremum of a properly normalized process belongs to the same space as the process itself. We also obtain estimates for the norm of this supremum. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 918–930, July, 1999.     

On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks

For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution of the supremum, infimum, and value of the process. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1359–1384, October, 2005.     

Hybrid simulation of space plasmas: Models with massless fluid representation of electrons. I. Collisionless shocks

This is a review of studies on hybrid simulation of low-frequency processes in space plasmas. We discuss the main approximations used in the derivation fo the hybrid model: particle representation for ions; massless fluid representation for electrons. The main numerical schemes for the implementation of this model are considered: the generalized Ohm law scheme, the predictor-corrector scheme, the scheme using Boris and Runge-Kutta methods to compute the fields. The article reviews the literature on imulation of collisionless shocks: quasiperpendicular shocks with anisotropic (mirror and ioncyclotron) instabilities; quasiparallel shocks with inclusion of re-formation processes (“periodic” destruction and repeated formation of the shock front), as well as collision of two shocks. Numerical aspects of simulation are discussed in some cases. Initialization of shocks and collisionless discontinuities is examined. Translated from Prikladnaya Matematika i Informatika, Vol. 11, No. 1, pp. 20–50, 1999.     

Nice local-global fields. I

The class NLGF of nice local-global fields and its three subclasses are introduced. The classes are stated to be recursively axiomatizable, and we establish a criterion of being elementarily embeddable for fields in mN LGF, fNLGF, and uNLGF. As a consequence, an elementary theory for the field of totally M-adic numbers is proved decidable for any finite set M of primes. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 411–423, July–August, 1996.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

Diffusions on path and loop spaces: existence, finite dimensional approximation and Hölder continuity

Summary. We construct Ornstein–Uhlenbeck processes and more general diffusion processes on path and loop spaces of Riemannian manifolds by finite dimensional approximation. We also show H?lder continuity of the sample paths w.r.t. the supremum norm. The proofs are based on the Lyons–Zheng decomposition. Received: 6 September 1996 / In revised form: 1 April 1997     

Lattices embeddable in subsemigroup lattices. I. Semilattices

V. B. Repnitskii showed that any lattice embeds in some subsemilattice lattice. In his proof, use was made of a result by D. Bredikhin and B. Schein, stating that any lattice embeds in the suborder lattice of a suitable partial order. We present a direct proof of Repnitskii’s result, which is independent of Bredikhin—Schein’s, giving the answer to a question posed by L. N. Shevrin and A. J. Ovsyannikov. We also show that a finite lattice is lower bounded iff it is isomorphic to the lattice of subsemilattices of a finite semilattice that are closed under a distributive quasiorder. Supported by INTAS grant No. 03-51-4110; RF Ministry of Education grant No. E02-1.0-32; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1; a grant from the Russian Science Support Foundation; SB RAS Young Researchers Support project No. 11. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 215–230, March–April, 2006.     

Strong Limit Theorems for Increments of Sums of Independent Random Variables

We derive universal strong laws for increments of sums of independent, nonidentically distributed, random variables. These results generalize universal results of the author for the i.i.d. case which include the strong law of large numbers, law of the iterated logarithm, Erdos-Renyi law, and Csorgo-Revesz laws. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 260–285.     

Toeplitz condition numbers as an <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> interpolation problem

The condition numbers CN(T) = ∥T∥ · ∥T⁻¹∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with ∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λ_i| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem. Bibliography: 2 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179.     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

Nonlinear 0–1 programming: I. Linearization techniques

Any real-valued nonlinear function in 0–1 variables can be rewritten as a multilinear function. We discuss classes of lower and upper bounding linear expressions for multilinear functions in 0–1 variables. For any multilinear inequality in 0–1 variables, we define an equivalent family of linear inequalities. This family contains the well-known system of generalized covering inequalities, as well as other linear equivalents of the multilinear inequality that are more compact, i.e., of smaller cardinality. In a companion paper [7]. we discuss dominance relations between various linear equivalents of a multilinear inequality, and describe a class of algorithms for multilinear 0–1 programming based on these results. Research supported by the National Science Foundation under grant ECS7902506 and by the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.     

A test for a hypothesis on the correlation function of Gaussian random processes

A test is proposed for a hypothesis on the correlation function of general Gaussian random processes. The test is based on theorems on estimates of the distribution of the supremum of sample estimators of correlation functions of Gaussian processes. For a wide class of stationary processes formulas are given that allow the test to be used immediately. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 61–74.     

Moments of the maximum of normed partial sums of random variables with multidimensional indices

Summary For a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm.     

Degrees of presentability of structures. I

Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of Σ-definability are the main object of investigation. It is shown that the semilattice of degrees of Σ-definability of countable structures agrees well with semilattices of T-and e-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations. Supported by RFBR grant Nos. 05-0100481 and 06-0104002, by the Council for Grants (under RF President) for State Support of Young Candidates of Science and Their Supervisors via project MK-1239.2005.1, and via INTAS project YSF 04-83-3310. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 763–788, November–December, 2007.     

Bose gas of anharmonic oscillators and refinement of the ZIPF law

We present a logarithmic law that passes into the Zipf law in the limit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 495–496, September, 2006.     

Generating triples of involutions for lie-type groups over a finite field of odd characteristic. I

It is proved that a simple Lie-type group of rank l≤4 over a field of odd characteristic is generated by three involutions of which two are commuting. As a consequence, the following results obtains: G is generated by two elements one of which is an involution and the order of the other is at most 2h, where h is the Coxeter number of a root system associated with G. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 77–96, January–February, 1997.     

Self-Normalized LIL for Hanson–Russo Type Increments

Let X ₁, X ₂,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.