[Paper Review] Product Formula, Independence and Asymptotic Moment-Independence for Complex Multiple Wiener-Ito Integrals

Yong Chen|arXiv (Cornell University)|Dec 5, 2014

Algebraic and Geometric Analysis10 references3 citations

TL;DR

This paper establishes a product formula for complex multiple Wiener-Itô integrals, enabling the derivation of independence and asymptotic moment-independence criteria. It proves the Üstünel-Zakai criterion for independence and extends the Nourdin-Rosiński asymptotic moment-independence framework to complex chaos, using connections with complex Hermite polynomials and symmetrized kernels.

ABSTRACT

We present the product formula for complex multiple Wiener-Ito integrals. As applications, we show the Ustunel-Zakai independent criterion, the Nourdin-Rosinski asymptotic moment-independent criterion and joint convergence criterion for complex multiple Wiener-Ito integrals.

Motivation & Objective

• To establish a product formula for complex multiple Wiener-Itô integrals, which was previously unknown.
• To derive a necessary and sufficient condition for independence of two complex multiple Wiener-Itô integrals using the Üstünel-Zakai criterion.
• To extend the Nourdin-Rosiński asymptotic moment-independence criterion to complex multiple Wiener-Itô integrals in the context of multiple chaos.
• To provide a theoretical foundation for joint convergence and moment-independent limit behavior in complex Wiener chaos.
• To leverage the relationship between real and complex Wiener-Itô integrals to derive related results in the complex setting.

Proposed method

• Uses the relationship between complex multiple Wiener-Itô integrals and complex Hermite polynomials as established by Itô [7] to derive the product formula.
• Employs symmetrization of kernels via the operator $\tilde{f}$ defined in equation (2.1), distinct from standard symmetrization in real chaos.
• Applies Itô's isometry and the norm inequality $\|\tilde{f}\| \leq \|f\|$ to ensure consistency and boundedness of integrals.
• Utilizes the connection between real and complex Wiener-Itô integrals (Theorem 3.3 of [3]) to extend known results to the complex case.
• Applies the Nourdin-Rosiński framework by analyzing covariances of squared magnitudes $\mathrm{Cov}(|F_{i,n}|^2, |F_{j,n}|^2)$ and their equivalence to vanishing contraction norms.
• Employs the contraction norm $\|f_{i,n} \otimes_{r,s} f_{j,n}\|$ and $\|f_{i,n} \otimes_{r,s} h_{j,n}\|$ for $r \leq a_i \wedge b_j$, $s \leq a_j \wedge b_i$, $r+s > 0$, to characterize asymptotic independence.

Experimental results

Research questions

• RQ1 What is the product formula for complex multiple Wiener-Itô integrals, and how does it differ from the real case?
• RQ2 Under what conditions are two complex multiple Wiener-Itô integrals independent?
• RQ3 What is the asymptotic moment-independence criterion for sequences of complex multiple Wiener-Itô integrals in multiple chaos?
• RQ4 How can the Nourdin-Rosiński joint convergence criterion be adapted to the complex Wiener-Itô setting?
• RQ5 What role do complex Hermite polynomials and symmetrized kernels play in deriving these results?

Key findings

• The paper establishes a product formula for complex multiple Wiener-Itô integrals via the symmetrization $\tilde{f}$ and complex Hermite polynomial identities.
• The Üstünel-Zakai independence criterion is extended to complex chaos: $I_{a,b}(f)$ and $I_{c,d}(g)$ are independent if and only if $\tilde{f} \otimes_{r,s} \tilde{g} = 0$ for all relevant contractions.
• The Nourdin-Rosiński asymptotic moment-independence criterion is generalized: asymptotic moment-independence holds if and only if $\lim_{n\to\infty} \|f_{i,n} \otimes_{r,s} f_{j,n}\| = 0$ and $\lim_{n\to\infty} \|f_{i,n} \otimes_{r,s} h_{j,n}\| = 0$ for all $i \neq j$ and valid $r,s$.
• Joint convergence of $d$-dimensional vectors of complex multiple Wiener-Itô integrals holds if the limit laws are moment-determined and the asymptotic moment-independence condition is satisfied.
• The criterion for moment-independence of limit laws is equivalent to vanishing cross-covariance $\mathrm{Cov}(|F_{i,n}|^2, |F_{j,n}|^2) \to 0$ for $i \neq j$, under moment-determinacy.
• The paper provides a full characterization of asymptotic moment-independence in complex chaos, including equivalence between covariance decay and contraction norm decay.