Multiple Wiener-Ito Integrals Research Articles

On almost sure convergence of random variables with finite chaos decomposition

Under mild conditions on a family of independent random variables $(X_{n})$ we prove that almost sure convergence of a sequence of tetrahedral polynomial chaoses of uniformly bounded degrees in the variables $(X_{n})$ implies the almost sure convergence of their homogeneous parts. This generalizes a recent result due to Poly and Zheng obtained under stronger integrability conditions. In particular for i.i.d. sequences we provide a simple necessary and sufficient condition for this property to hold. We also discuss similar phenomena for sums of multiple Wiener-Ito integrals with respect to Poisson processes, answering a question by Poly and Zheng.

Fractional Kinetic Equation Driven by General Space–Time Homogeneous Gaussian Noise

In this article, we consider a class of fractional kinetic equation driven by a space–time homogeneous Gaussian noise on $$[0,T]\times {\mathbb {R}}^d$$ with $$T>0$$ . Under some mild assumptions on the spatial spectral measure of the Gaussian noise, we prove the existence of mild solution of this equation (in the Skorohod sense) and the Holder continuity of its sample paths. Our techniques are based on multiple Wiener-Ito integral and Malliavin calculus.

Strong asymptotic independence on Wiener chaos

Let $F_n = (F_{1,n}, ....,F_{d,n})$, $n\geq 1$, be a sequence of random vectors such that, for every $j=1,...,d$, the random variable $F_{j,n}$ belongs to a fixed Wiener chaos of a Gaussian field. We show that, as $n\to\infty$, the components of $F_n$ are asymptotically independent if and only if ${\rm Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $i\neq j$. Our findings are based on a novel inequality for vectors of multiple Wiener-Ito integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosinski.

Asymptotic properties of U-processes under long-range dependence

Let $(X_i)_{i\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k)=\PE(X_{1}X_{k+1})$ satisfying: $\rho(0)=1$ and $\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\{U_n(r),\; r\in I\}$ defined as $$ U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j\leq n}\1_{\{G(X_i,X_j)\leq r\}}\; , $$ where $I$ is an interval included in $\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated scale estimator. The limiting distributions are expressed through multiple Wiener-Ito integrals.

Schoenberg’s Theorem and Unitarily Invariant Random Arrays

Motivated by Schoenberg’s theorem in classical analysis and some recent work on the circular law, we extend the basic representations of unitarily invariant random arrays and functionals to the complex case. As in the real case, the basic building blocks are multiple Wiener–Ito integrals on tensor products of a separable Hilbert space. The complex setting leads to some unexpected simplifications.

Stein’s method and Normal approximation of Poisson functionals

We combine Stein’s method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry–Esseen bounds in Central limit theorems (CLTs) involving multiple Wiener–Ito integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein–Uhlenbeck Levy processes.

Stein’s method on Wiener chaos

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Ito integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esseen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.

On a Multivariate Version of Bernstein's Inequality

We prove such a multivariate version of Bernstein's inequality about the tail distribution of degenerate $U$-statistics which is an improvement of some former results. This estimate will be compared with an analogous bound about the tail distribution of multiple Wiener-Ito integrals. Their comparison shows that our estimate is sharp. The proof is based on good estimates about high moments of degenerate $U$-statistics. They are obtained by means of a diagram formula which enables us to express the product of degenerate $U$-statistics as the sum of such expressions.

Gaussian Approximations of Multiple Integrals

Fix $k\geq 1$, and let $I(l), l \geq 1$, be a sequence of $k$-dimensional vectors of multiple Wiener-Ito integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as $l \to\infty$, the law of $I(l)$ is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a $k$-dimensional Gaussian vector having the same covariance matrix as $I(l)$. The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of $I(l)$. In particular, we will not assume that the covariance matrix of $I(l)$ is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.

Tail behaviour of multiple random integrals and U-statistics

This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of U-statistics and multiple Wiener–Ito integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or U-statistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a non-expert also can understand, and not only on a formal level.

General Autoregressive Models with Long-Memory Noise

We give the limiting distribution of the least-squares estimator in the general autoregressive model driven by a long-memory process. We prove that with an appropriate normalization the estimation error converges, in distribution, to a random vector which contains: (1) a stochastic component, due to the presence of the unstable roots, which are multiple Wiener–Ito integrals and a non-linear functionals of stochastic integrals with respect to a Brownian motion; (2) a constant component due to the stable roots; (3) a stochastic component, due to the presence of the explosive roots, which is a mixture of normal distributions.

Two results on multiple Stratonovich integrals

Formulae connecting the multiple Stratonovich integrals with single Ogawa and Stratonovich integrals are derived. Multiple Riemann-Stieltjes integrals with respect to certain smooth approximations of the Wiener process are considered and it is shown that these integrals converge to multiple Stratonovich integrals as the approximation converges to the Wiener process. In an important work Hu and Meyer (1987) introduced a new multi- ple stochastic integral with respect to a Wiener process, called the multiple Startonovich integral (MSI), which is in general different from the usually studied multiple Wiener-Ito integral. However, Hu and Meyer only offered some rather informal definitions and proofs. Johnson and Kallianpur (1993) gave a rigor- ous definition for the MSI (the term MSI was not used in their work), denoted in this work by δp(·), and gave necessary and sufficient conditions for its exis- tence. Recently, multiple Stratonovich integrals have been applied to problems in asymptotic statistics and nonlinear filtering (cf. Budhiraja and Kallianpur (1995, 1996)). In this work we study some properties of multiple Stratonovich integrals which also give some justification for the appearance of the name 'Stratonovich' in the integral. We recall that one of the important properties of multiple Wiener-Itoi nte- grals of symmetric kernels is that they can be expressed as iterated (indefinite) Ito-integrals, by which we mean that if fp ∈ L 2(0, 1) p (the class of real val- ued, square integrable, symmetric functions defined on (0, 1) p ) then the multiple

Multiple Intersection Local Time of Planar Brownian Motion as a Particular Hida Distribution

Subspaces of the space of Hida distributions are quantified in which the multiple intersection local time of planar Brownian motion, renormalized by taking off the term of order zero in its chaos decomposition, lives. This is done by means of a new formula expressing the expectation of higher products of multiple Wiener–Ito-integrals by certain functions of the scalar products of their kernels.

Chaos expansions of double intersection local time of Brownian motion in [formula omitted] and renormalization

Double intersection local times α( x,.) of Brownian motion W if R d which measure the size of the set of time pairs ( s, t), s ≠ t, for which W t and W s + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of α( x,.) in terms of eventually negative order Sobolev spaces with respect to the canonical Dirichlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of α( x,.) as | x| → 0. The results, valid for any dimension d, describe a pattern in which the well known cases d = 2, 3 are naturally embedded.

Random arrays and functionals with multivariate rotational symmetries

Our aim is to extend Schoenberg's classical theorem to higher dimensions, by establishing representations of arbitrary separately or jointly rotatable continuous linear random functionals in terms of multiple Wiener-Ito integrals and their tensor products. This leads to similar representations for separately or jointly rotatable arrays, and for separately or jointly exchangeable or spreadable random sheets.

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

LetL(x, T),x∈Rd,T∈R+N, be the local time of theN-parameter Wiener processW taking values inRd. Even in the distribution valued casedd≧2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|→0 and/orT∞ and of related occupation integrals\(X_T (f) = \int\limits_{[0,T]} f (W_S )\) asT→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.XT(f), and of the second order, i.e. normalized convergence laws forL(x, T)−E(L(x, T)) resp.XT(f)−E(XT(f)).

On theL2-Theory of product stochastic measures and multiple wiener–ito integrals

In this work we construct product stochastic measures using symmetric tensor products in appropriate Hilbert spaces and then construct multiple Wiener-Itô integrals using the vector valued measure approach. A clear relationship between the theories of L 2 -multiple stochastic integrals and vector valued measures is established. This approach yields, as special cases, the multiple stochastic integrals of Itô Surgailis, Ogura, and Fox and Taqqu, among others

Resampling U-statistics using p-stable laws

It is well known that symmetric statistics based on a kernel with finite second moment have a limit law which can be described by a multiple Wiener-Ito integral. However, if the kernel has less than second moments, no weak limit law holds in general. In the present paper we show that by a suitable change of the empirical process this process has a p-stable multiple integral as its limit.

Multiple Wiener-Ito integrals possessing a continuous extension

LetF(W) be a Wiener functional defined byF(W)=I n(f) whereI n(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL 2([0, 1] n ) kernelf. We show that a necessary and sufficient condition for the existence of a continuous extension ofF, i.e. the existence of a function o(·) from the continuous functions on [0, 1] which are zero at zero to ℝ which is continuous in the supremum norms and for which o(W)=F(W) a.s, is that there exists a multimeasure μ(dt 1,...,dt n ) on [0, 1] n such thatf(t 1, ...,t n ) = μ((t 1, 1]), ..., (t n , 1]) a.e. Lebesgue on [0, 1] n . Recall that a multimeasure μ(A 1,...,A n ) is for every fixedi and every fixedA i,...,Ai-1, Ai+1,...,An a signed measure inA i and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t 1,t 2, ...,t n ) = μ((t 1, 1], ..., (t n , 1]) then all the tracesf (k), $$k \leqq \left[ {\frac{n}{2}} \right]$$ off exist, eachf(k) induces ann−2k multimeasure denoted by μ(k), the following relation holds $$I_n (f) = \sum\limits_{k = 0}^{[n/2]} {\left( { - \frac{1}{2}} \right)^k \frac{{n!}}{{k!(n - 2k)!}}\int\limits_{[0,1]^{n - 2k} } {W_{t_1 } } \cdot \cdot \cdot W_{t_{n - 2k} } \cdot \mu ^{(k)} (dt_1 ,...,dt_{n - 2k} )} $$ and each of the integrals in the above expression equals the multiple Stratonovich or Ogawa type integral of the tracef(k), namely $$\int\limits_{[0, 1]^{n - 2k} } {W_{t_1 } } \cdot \cdot \cdot W_{t_{n - 2k} } \mu ^{(k)} (dt_1 , . . . , dt_{n - 2k} ) = I_{n - 2k} \circ (f^{(k)} ).$$

On Independence and Conditioning On Wiener Space

Let $I_p(f)$ and $I_q(g)$ be multiple Wiener-Ito integrals of order $p$ and $q$, respectively. A characterization of independence of general random variables on Wiener space in the context of the stochastic calculus of variations is derived and a necessary and sufficient condition on the pair of kernels $(f, g)$ is derived under which the random variables $I_p(f), I_q(g)$ are independent.