Wiener-Ito Integrals Research Articles

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.

On the moments of a multiple wiener-ito integral and the space induced by the polynomials of the integral

In the first part of this note we derive a lower bound for E\Ip(f)\m where Ip(f) is the multiple Wiener-Ito integral of the kernel f(t1,-.,tp), t∈T and T=[0, ∈). In the second part we consider the linear space H{Ip(f)) of the L2 functionals induced by the random variables {(Ip(f))m,meN}. For ≥ it is not known whether H{Ipf} is the same space of L2 functionals which are measurable with respect to the subsigma field induced by the random variable Ip(f). In the final part of this note some special results concerning the conditional expectation on the Wiener space are derived. In particular, it is shown that the conditional expectation of a random variable belonging to an odd chaos given the even chaos is zero.

A survey of functional laws of the iterated logarithm for self-similar processes

A process X(t) is self-similar with index H > 0 if the finite-dimensional distributions of X(at) are identical to those of aHX(t) for all a > 0. Consider self-similar processes X(t) that are Gaussian or that can be represented throught Wiener-Ito integrals. The paper surveys functional laws of the iterated logarithm for such processes X(t) and for sequences whose normalized sums coverage weakly to X(t). The goal is to motivate the results by including outline of proofs and by highlighting relationships between the various assumptions. The paper starts with a general discussion fo functional laws of the iterated logarithm, states some of their formulations and sketches the reproducing kernal Hilbert space set-up.

Strongly correlated random fields as observed by a random walker

We are given a random walk S 1, S 2, ... on ℤν, ν≧1, and a strongly correlated stationary random field ξ(x), xeℤν, which is independent of the random walk. We consider the field as observed by a random walker and study partial sums of the form \(W_n = \sum\limits_{j = {\text{ }}1}^n {\xi (S_j )}\). It is assumed that the law corresponding to the random walk belongs to the domain of attraction of a non-degenerate stable law of index β, 0<β≦2. We further suppose that the field ξ satisfies the non-central limit theorem of Dobrushin and Major with a scaling factor \(n^{ - v + \tfrac{1}{2}\alpha k} ,\alpha k < v\). Under the assumption αk<β it is shown that \(n^{ - 1 + \tfrac{1}{2}\alpha k/\beta } {\text{ }}W_{[nt]} \) converges weakly as n→∞ to a self-similar process {Δ t , t≧0} with stationary increments, and Δ t can be represented as a multiple Wiener-Ito integral of a random function. This extends the noncentral limit theorem of Dobrushin and Major and yields a new example of a self-similar process with stationary increments.