On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Weighted approximation of random functions

ＷＥＩＧＨＴＥＤＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＲＡＮＤＯＭＦＵＮＣＴＩＯＮＳＹＵＪＩＡＲＯＮＧＡｂｓｔｒａｃｔ：Ｌｅｔ（Ω,Ａ,Ｐ）ｂｅａｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ,Ｘ（ｔ,ω）ａｒａｎｄｏｍｆｕｎｃｔｉｏｎｃｏｎｔｉｎｕｏｕｓｉｎｐｒｏｂａｂ...     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

On driftless one-dimensional SDE’s with respect to stable Levy processes

The time-dependent SDE dX _t = b(t, X _t−)dZ _t with X ₀ = x ₀ ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation in class of time change processes, where is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 517–531, October–December, 2007.     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

On a semigroup generated by a convex combination of two Feller generators

Let be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in with related Feller processes {X _A (t), t ≥ 0} and {X _B (t), t ≥ 0} and let α and β be two non-negative continuous functions on with α + β = 1. Assume that the closure C of C ₀ = αA + βB with generates a Feller semigroup {T _C (t), t ≥ 0} in . It is natural to think of a related Feller process {X _C (t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point , with probability α(p) the process behaves like {X _A (t), t ≥ 0} and with probability β(p) it behaves like {X _B (t), t ≥ 0}. We provide an approximation of {T _C (t), t ≥ 0} via a sequence of semigroups acting in that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].     

r-Maximal major subsets

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ^*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^*(A) and ℒ^*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_rm A as those with a Δ₃ function that for each recursiveR _i specifiesR _i or as infinite on and to be preferred in the construction ofB. There are r.e.A andB with ℒ^*(A) and ℒ^*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^*,A〉 and 〈ℰ^*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies . The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

Large and Moderate Deviations for Estimators of Quadratic Variational Processes of Diffusions

For a diffusion process dX_t = σdB _t + b(t, X_t)dt with (σ _t ) unknown, we study the large and moderate deviations of the estimator of the quadratic variational process . This revised version was published online in June 2006 with corrections to the Cover Date.     

Spectral Conditions for Strong Local Nondeterminism and Exact Hausdorff Measure of Ranges of Gaussian Random Fields

Let X={X(t),t∈ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

On the spectral representation of the elementary operator and spectral mapping theorems

LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X) ⁿ ,B∈B(Y) ⁿ , the elementary operator acting onB(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS _p (L _A ,R _B )=σ(A)×σ(B) and .     

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB ^m ×C ⁿ be a compact set over the unit sphere ϱB ^m such that for eachz∈ϱB ^m the fiberX _z ={ω∈C ⁿ ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC ⁿ . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z ₀,w ₀)∈Int there exists a smooth up to the boundary analytic discF:Δ→B ^m ×C ⁿ with the boundary inX such thatF(0)=(z ₀,w ₀). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.     

Density estimation for a class of stationary nonlinear processes

Let {X _t ;t∈ℤ be a strictly stationary nonlinear process of the formX _t =ε _t +∑ _r=1 ^∞ W _rt , whereW _rt can be written as a functiong _r (ε _t−1,...ε _t-r-q ), {ε _t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁| ^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg _r and {ε _t } we then show thatX ₁ has a density functionf and that the standard kernel type estimator baded on a realization {X ₁,...,X _n } from {X _t } is, asymptotically, normal and converges a.s. tof(x) asn→∞. The research of this author was partially carried out while he was a research scholar, on a sabbatical leave, at the Department of Statistics and Probability, Michigan State University.     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Asymptotic Independence of Distant Partial Sums of Linear Processes

We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) for τ ∈ [0, 1], where and are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X _t , t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We consider the cases where (X _t ) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency of certain increment-type statistics in moving-average observations. This work supported by the joint Lithuania-French research program Gilibert. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005.     

A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.