Weak invariance principle for finite-populationU-statistics

Under the weakest possible conditions, we establish the weak invariance principle for finite-populationU-statistics in this paper. It is worth while to point out that, for the sampling without replacement, the sequence of random delements inC[0, 1], associated with the sample partial sums or theU-statistics, converges in law to the standard Brown bridge, but not to the Brown motion as in the usual case of replacement sampling.     

Weak invariance principle for local times

Results are found on weak convergence of some processes generated by a recurrent random walk to a Brownian local time process.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 14–31, 1987.     

A generalized skewness statistic for stationary ergodic martingale differences

We present a class of generalized skewness statistics depending on a parameter β > 0 and containing the usual skewness statistic when β = 3, but providing greater flexibility for modelling and testing skewness when β ≠ 3. The statistics’ suitability for financial applications is illustrated using a large data set from the Australian share market. Data is assumed to be observations on stationary ergodicmartingale differences with possibly leptokurtic marginals, rather than independent identically distributed samples. The statistics can be studentized for use in hypothesis testing. Proof is provided of their asymptotic distributions undermild assumptions. Rates of convergence and power of the tests against skewed alternatives are assessed using simulation.     

Weak martingale Hardy spaces and weak atomic decompositions

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale Hp-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak Lp-norm, the inequalities of weak (p ,p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale Hp-theory.     

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.⁽⁷⁾ Then we use the chaining argument with stratification adapted from that of Ossiander.⁽¹¹⁾ The results of this paper generalize those of Levental⁽¹⁰⁾ and Ossiander.⁽¹¹⁾     

Cesaro mean convergence of martingale differences in rearrangement invariant spaces

We study the class of r.i. spaces in which Cesaro means of any weakly null martingale difference sequence is strongly null. This property is related to the Banach-Saks property. We show that in classical (separable) r.i. spaces (such as Orlicz, Lorentz and Marcinkiewicz spaces) these properties coincide but this is no longer true for general r.i. spaces. We locate also a class of r.i. spaces having this property where an analogue of the classical Dunford-Pettis characterization of relatively weakly compact subsets in L ₁ holds. Research was partially supported by the ARC and NSF grant DMS-0244515.     

B-Products of Besov type spaces and some applications

The paper derives a representation of classical Besov zero spaces by means of the so called B-products of the generalized Besov type zero spaces, generated by completely regular polyhedrons. Peetre K-functional is estimated and an interpolation formula of “real method” is proved for pairs of B-products. Some applications of these results are given.     

Weak convergence to the fractional Brownian sheet using martingale differences

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane.     

Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

We study the weak convergence of the family of processes {V _n (t)} _n??? defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {?? _n (u)} _n??? is a family of processes converging in law to a Brownian motion, as n????. We consider two cases of {?? _n }. First, we construct ?? _n based on the well-known Donsker??s theorem and show that {V _n (t)} _n??? converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n????. Second, we construct ?? _n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {V _n (t)} _n???.     

An invariance principle for stationary random fields under Hannan’s condition

We establish an invariance principle for a general class of stationary random fields indexed by Z^d${\mathbb{Z}}^d$, under Hannan’s condition generalized to Z^d${\mathbb{Z}}^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.     

Weak invariance principle for solutions of stochastic recurrence equations in a banach space

We show that, in a Banach space, continuous random processes constructed by using solutions of the difference equationX _n =A _n X _n+1+V _n , n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 114–117, January, 1995.     

Characterizations of realized homogeneous Besov and Triebel-Lizorkin spaces via differences

Based on the role of the polynomial functions on the homogeneous Besov spaces, on the homogeneous Triebel-Lizorkin spaces and on their realized versions, we study and obtain characterizations of these spaces via difference operators in a certain sense.     

Embeddings for anisotropic Besov spaces

We prove embedding theorems for fully anisotropic Besov spaces. In particular, inequalities between modulus of continuity in different metrics and of Sobolev type are obtained. Our goal is to get sharp estimates for some anisotropic cases previously unconsidered.     

Interpolation theorems for Lorentz martingale spaces

We apply function parameters, which was introduced by T.?F. Kalugina, to interpolation between martingale Hardy spaces. Some new interpolation theorems for Lorentz martingale spaces are proved.     

Boundedness of the Hilbert transformation in some weighted Besov type spaces

The Hilbert transformation is studied on a new weighted Besov type space, characterized by means of the best approximation. This space appears as a limit case of the well-known Besov spaces defined by Ditzian and Totik, and it can be described with the help ofK-functionals and -moduli of smoothness.Work supported by Vigoni Project 1996/97 (Prot.Am/301-96/P/RG)