Complex Wiener-Itô Chaos Decomposition Revisited

Chen, Yong; Liu, Yong

doi:10.1007/s10473-019-0311-6

Cited by 5 publications

(2 citation statements)

References 35 publications

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“…In this section, we briefly introduce some basic theories of the isonormal Gaussian process, Malliavin calculus, cumulants and multidimensional Stein's method. See [7,15,22,25] for more details.…”

Section: Preliminariesmentioning

confidence: 99%

Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral

Chen¹

2023

Preprint

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We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Itô integrals of which components all belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's method for normal approximation and method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergences for complex Wiener-Itô integrals, vector-valued Wiener-Itô integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals.

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Section: Preliminariesmentioning

confidence: 99%

Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral

Chen¹

2023

Preprint

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“…Complex Hermite polynomials were discovered by [35] in the context of complex multiple Wiener integrals. They share many properties of standard Hermite polynomials [14] [34] [27]. To prove Theorem 1.2 we shall employ a so-called Wiener chaos expansion in complex Hermite polynomials, which is in many ways analogous to the more standard chaos expansion in real Hermite polynomials.…”

Section: Chaos Expansion On the Complex Planementioning

confidence: 99%

Untitled

2020

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We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set).We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so.Our work contributes to recent developments in statistical signal processing, where the timefrequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram ("silent points"). We show that empirical spectrogram zero counts enjoy moderate deviation from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectogram maxima ("loud points") fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners.

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Kernel representation formula: From complex to real Wiener–Itô integrals and vice versa

Chen,

Liu

2024

Stochastic Processes and their Applications

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Complex Wiener-Itô Chaos Decomposition Revisited

Cited by 5 publications

References 35 publications

Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral

Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral

Untitled

Kernel representation formula: From complex to real Wiener–Itô integrals and vice versa

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