The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

An explicit Lévy-Hinčin formula for convolution semigroups on locally compact groups

LetG be a locally compact group and (_t)_{t 0} a continuous convolution semigroup of probability measures onG. We show that an operatorN is the infinitesimal generator of (_t)_{t 0} iffN is defined at least on the spaceC ₂(G) of twice right differentiable functions and if

Limit theorems for transient diffusions on the line

Summary LetX be a diffusion in natural scale on (0,1], with 1 reflecting, and letc(x)(H _x ) andv(x)var (H _x ), whereH _x =inf{t: X _t =x}. Let _x =sup{t:X _t =x}. The main results of this paper are firstly that (i)c is slowly varying; (ii) are all equivalent: and secondly that (v) are all equivalent, and are implied by the condition . Other partial results for more general limit theorems are proved, and new results on regular variation are established.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Birth delay of a Markov process and the stopping distributions for regular processes

Summary We consider Markov processes with a fixed transition functionp(r, x; t, B) and with random birth times. We show that a process can be obtained from (X _t ,P) by birth delay if and only if for allt andB. As an application, we give a new version and a new proof of the results of Rost [R] and Fitzsimmons [F2] on stopping distributions of Markov processes. The key Lemma 1.1 replaces the filling scheme used by the previous authors.Birth delay was considered from a different prospective in [F1].Partially supported by the National Science Foundation Grant DMS-8802667     

A new proof of the Komlós-Révész-theorem

The Komlós-Révész theorem states: For r.v.s.X _n with X _{n 1}M there exists a subsequenceX _{k n} and a r.v.X with X₁M such that

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

On The Invariant Measure of a Positive Recurrent Diffusion in $${\mathbb{R}}$$

Given a one-dimensional positive recurrent diffusion governed by the Stratonovich SDE , we show that the associated stochastic flow of diffeomorphisms focuses as fast as , where is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is . Applications to stationary solutions of X _t , asymptotic behavior of solutions of SPDEs and random attractors are offered. This paper was written while the author was visiting Northwestern University and the opinions expressed in it are those of the author alone and do not necessarily reflect the views of Merrill Lynch, its subsidiaries or affiliates.     

Some Processes Associated with Fractional Bessel Processes

Let be a d-dimensional fractional Brownian motion with Hurst parameter H and let be the fractional Bessel process. Itôs formula for the fractional Brownian motion leads to the equation . In the Brownian motion case is a Brownian motion. In this paper it is shown that X_t is not an -fractional Brownian motion if H 1/2. We will study some other properties of this stochastic process as well.     

On the limiting behaviour of Lévy processes at zero

We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to and as where X is a Lévy Process and T _r the time it first leaves an interval of radius r     

Central limit theorems for nonlinear functionals of stationary Gaussian processes

Let X=(X _t ,t) be a stationary Gaussian process on (, ,P), letH(X) be the Hilbert space of variables inL ² (,P) which are measurable with respect toX, and let (U _s ,s) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

On Measure-Valued Processes Generated by Differential Equations

We study the problem of representation of a homogeneous semigroup { _t } _{t 0} of transformations of probability measures on in the form where satisfies a differential equation of a special form dependent on the measure . We give necessary and sufficient conditions for this representation.     

A new product of algebras and a type reduction theorem

A construction is defined which associates, to every algebra of a fixed but arbitrary finite similarity type, a groupoidF . The identities ofF are finitely based if and only if those of are, andF is finite if and only if is finite. Up to isomorphism,F has the same endomorphism monoid and subalgebra lattice as , but the congruence lattice ofF is the result of adjoining a new 1 to the congruence lattice of .F is functorial, preserves the satisfaction (and the non-satisfaction) of most Mal'cev conditions, and produces, by composition with the operation of forming the generated variety, an isomorphism of the lattice of varieties of fixed type to an interval in the lattice of varieties of groupoids.The construction makes use of a new product operation, applicable to two algebras of differing similarity types, which is introduced and studied in this paper.Research supported by National Science Foundation grant MCS-8103455.Presented by K. A. Baker.     

Remarks on Estimation Problem for Stationary Processes in Continuous Time

The problem of estimating of the law (in the space of the paths) and the common marginal distribution for a strictly stationary ergodic process X is discussed. We show, in particular, that:(1) The empirical measure

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).     

Convergence of Sums of Appell Polynomials with Infinite Variance

We investigate the convergence of distributions of partial sums of Appell polynomials of a long-memory moving average process X _t with i.i.d. innovations _s in the case where the variance , and the distribution of #x03BE; ₀ ^m belongs to the domain of attraction of an -stable law with 1<< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an -stable Lévy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of , m, and d, where 0X _t.     

First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Let (B _t) _t0 be standard Brownian motion starting at y, X _t = x + ^t ₀ V(B _s) ds for x (a, b), with V(y) = y if y0, V(y)=–K(–y) if y0, where >0 and K is a given positive constant. Set _ab=inf{t>0: X _t(a, b)} and ₀=inf{t>0: B _t=0}. In this paper we give several informations about the random variable _ab. We namely evaluate the moments of the random variables , and also show how to calculate the expectations . Then, we explicitly determine the probability laws of the random variables as well as the probability by means of special functions.     

Some numerical results on best uniform rational approximation ofx α on [0,1]

Let be a positive number, and letE _n,n (x ;[0,1]) denote the error of best uniform rational approximation from _n,n tox on the interval [0,1]. We rigorously determined the numbers {E _n,n (x ;[0,1])} _n ^=1/30 for six values of in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of , Richardson's extrapolation was applied to the products to obtain estimates of

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .