On the deterministic property for characteristic functions of several variables

TL;DR

This paper investigates conditions under which characteristic functions of probability measures can be uniquely extended from neighborhoods at infinity, highlighting the importance of support properties and arithmetic conditions of the measure's density.

Contribution

It provides sufficient conditions involving the support and density of the measure for the uniqueness of characteristic function extensions at infinity.

Findings

Conditions on the support and density ensure unique extension of characteristic functions.

The size and arithmetic properties of the support are crucial for extension uniqueness.

Optimality of the derived conditions is analyzed.

Abstract

Assume that $f$ is the characteristic function of a probability measure $\mu_f$ on $R^n$. Let $\sigma>0$. We study the following extrapolation problem: under what conditions on the neighborhood of infinity $V_{\sigma}=\{x\in R^n: |x_k|>\sigma, \ k=1,\dots, n\}$ in $R^n$ does there exist a characteristic function $g$ on $R^n$ such that $g=f$ on $V_{\sigma}$, but $g\not\equiv f$? Let $\mu_f$ have a nonzero absolutely continuous part with continuous density $\varphi$. In this paper certain sufficient conditions on $\varphi$ and $V_{\sigma}$ are given under which the latter question has an affirmative answer. We also address the optimality of these conditions. Our results indicate that not only does the size of both $V_{\sigma}$ and the support ${{\text{\,supp}}\,}\varphi$ matter, but also certain arithmetic properties of ${{\text{\,supp}}\,}\varphi$.