[Paper Review] Convergence of densities of some functionals of Gaussian processes

Yaozhong Hu, Fei Lu+1 more|arXiv (Cornell University)|Feb 27, 2013

Probability and Risk Models31 references7 citations

TL;DR

This paper establishes uniform convergence of densities of functionals of Gaussian processes to the normal density using Malliavin calculus and Stein's method. It provides explicit upper bounds for the uniform distance between the density of a multiple stochastic integral in the $q$th Wiener chaos and the normal density, under non-degeneracy conditions involving negative moments of the Malliavin derivative, with the bound proportional to $\sqrt{\mathbb{E}[F^4] - 3\sigma^4}$.

ABSTRACT

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein's method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed.

Motivation & Objective

• To establish uniform convergence of the densities of sequences of functionals of Gaussian processes to the normal density.
• To derive explicit quantitative bounds for the uniform distance between the density of a random variable in a fixed Wiener chaos and the normal density.
• To extend the analysis to higher-order derivatives of the density and to random vectors in Wiener chaos under Malliavin non-degeneracy.
• To apply the results to parameter estimation in Ornstein-Uhlenbeck processes, particularly the least squares estimator.

Proposed method

• Utilizes Malliavin calculus to express the density of a multiple stochastic integral $F = I_q(f)$ via integration by parts: $f_F(x) = \mathbb{E}[\mathbf{1}_{\{F>x\}} q\|DF\|_{\mathfrak{H}}^{-2}F] - \mathbb{E}[\mathbf{1}_{\{F>x\}} \langle DF, D(\|DF\|_{\mathfrak{H}}^{-2}) \rangle_{\mathfrak{H}}]$.
• Applies Stein's method to estimate the difference $|\mathbb{E}[\mathbf{1}_{\{F>x\}}F] - \mathbb{E}[\mathbf{1}_{\{N>x\}}N]|$, which controls the normal approximation error.
• Employs Hölder's inequality and moment estimates for iterated derivatives of $\|DF\|_{\mathfrak{H}}^{-2}$, assuming $\mathbb{E}[\|DF\|_{\mathfrak{H}}^{-6}] < \infty$.
• Derives bounds on the $m$th derivative of the density by requiring $\mathbb{E}[\|DF\|_{\mathfrak{H}}^{-\beta}] < \infty$ for $\beta > 6m + 6(\lfloor m/2 \rfloor \vee 1)$.
• Uses a compactness argument and Sobolev norm estimates for the inverse density $\gamma_F^{-1}$ to handle vector-valued functionals.
• Combines the integration-by-parts formula with Malliavin calculus identities involving $D_u^k \delta_u$ and $w = \|DF\|_{\mathfrak{H}}^2$ to control higher-order derivatives.

Experimental results

Research questions

• RQ1 Under what conditions does the density of a multiple stochastic integral in the $q$th Wiener chaos converge uniformly to the normal density?
• RQ2 Can explicit quantitative bounds be derived for the uniform distance between the density of a functional of a Gaussian process and the normal density?
• RQ3 How do the bounds depend on the fourth moment of the functional and the negative moments of the Malliavin derivative?
• RQ4 Can the convergence results be extended to higher-order derivatives of the density and to random vectors in Wiener chaos?
• RQ5 What are the implications of these bounds for the convergence of densities in statistical estimation problems, such as the least squares estimator in Ornstein-Uhlenbeck processes?

Key findings

• For a random variable $F = I_q(f)$ in the $q$th Wiener chaos with $\mathbb{E}[F^2] = \sigma^2$ and $\mathbb{E}[\|DF\|_{\mathfrak{H}}^{-6}] < \infty$, the uniform distance between its density $f_F$ and the normal density $\phi$ satisfies $\sup_x |f_F(x) - \phi(x)| \leq C \sqrt{\mathbb{E}[F^4] - 3\sigma^4}$, where $C$ depends on $q$, $\sigma$, and the sixth negative moment of $\|DF\|_{\mathfrak{H}}$.
• The bound can be equivalently expressed in terms of $\sqrt{\mathrm{Var}(\|DF\|_{\mathfrak{H}}^2)}$, linking the convergence to the variance of the Malliavin derivative norm.
• For the $m$th derivative of the density, uniform bounds are obtained under the condition $\mathbb{E}[\|DF\|_{\mathfrak{H}}^{-\beta}] < \infty$ with $\beta > 6m + 6(\lfloor m/2 \rfloor \vee 1)$, ensuring smoothness of the convergence.
• The results extend to random vectors in Wiener chaos that are uniformly non-degenerate in the Malliavin sense, with bounds derived via Sobolev norms of the inverse density.
• In the second Wiener chaos, explicit upper bounds for the uniform norm of the density difference are derived, providing a quantitative estimate for non-Gaussian functionals.
• An application to the least squares estimator of the drift parameter in an Ornstein-Uhlenbeck process is provided, showing that the density of the estimator converges uniformly to normality under the same moment conditions.