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Some Gaussianity criteria for Khinchin families

Víctor J. Maciá

Abstract

In this note we establish simple and verifiable analytical conditions for a power series f in the class K, and its associated Khinchin family, to be Gaussian. We give several moment criteria for general power series with non-negative coefficients in K and also provide several applications of these criteria.

Some Gaussianity criteria for Khinchin families

Abstract

In this note we establish simple and verifiable analytical conditions for a power series f in the class K, and its associated Khinchin family, to be Gaussian. We give several moment criteria for general power series with non-negative coefficients in K and also provide several applications of these criteria.
Paper Structure (23 sections, 23 theorems, 157 equations)

This paper contains 23 sections, 23 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Asymptotic notations.
  3. Khinchin families
  4. Mean and variance functions.
  5. Normalization and characteristic functions.
  6. Gaussian power series.
  7. Fulcrum of a power series.
  8. Zero-free region
  9. Characteristic function of the family $(\breve{X}_t)$ and the fulcrum.
  10. Expressions for the moments of $(\breve{X}_t)$ in terms of the fulcrum.
  11. Riordan formula.
  12. Moment criteria for Gaussianity
  13. A partial converse of Theorem \ref{['thm: gaussianity all derivatives']}.
  14. A partial characterization.
  15. Some applications
  16. ...and 8 more sections

Key Result

Proposition A

Let $f\in \mathcal{K}$ have radius of convergence $R>0$. If for some $t\in [0,R)$ and some $\theta \in [-\pi, \pi]$, we have $f(te^{i \theta})=0$, then Thus, $f\in\mathcal{K}$ does not vanish in

Theorems & Definitions (42)

  • Proposition A
  • Lemma 2.1
  • Theorem 1
  • proof
  • Corollary 2.1
  • proof
  • Remark 3
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 32 more