On the deterministic property for characteristic functions of several variables

Abstract

Assume that $f$ is the characteristic function of a probability measure $μ_f$ on $R^n$. Let $σ>0$. We study the following extrapolation problem: under what conditions on the neighborhood of infinity $V_σ=\{x\in R^n: |x_k|>σ, \ k=1,\dots, n\}$ in $R^n$ does there exist a characteristic function $g$ on $R^n$ such that $g=f$ on $V_σ$, but $g\not\equiv f$? Let $μ_f$ have a nonzero absolutely continuous part with continuous density $φ$. In this paper certain sufficient conditions on $φ$ and $V_σ$ 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 $V_σ$ and the support ${\text{\,supp}\,}φ$ matter, but also certain arithmetic properties of ${\text{\,supp}\,}φ$.