The problem of stability for J. Marcinkiewicz's theorem

N. A. Sapogov

Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 104-124

Résumé

In this paper we investigate stability of the well known theorem due to J. Marciniciewicz asserting that $\exp P(t)$ where $P(t)$ is a polynomial can be a characteristic function only when the degree of $P(t)$ is $\leq2$. Our main result is given by the following theorem. Theorem. {\it Let $|\exp P_{2n}(t)-\varphi(t)|\leq\varepsilon,\quad t\in[-T,T]$, where $$ P_{2n}(t)=-\frac12t^2+\sum_{k=2}^n a_{2k}t^{2k}, \quad a_{2k}\in R^1,\quad|a_{2k}|\leq H,\quad k=2,3,\dots,n,\quad a_{2n}<0 $$ $\varphi(t)=\varphi(-t)$ – even characteristic function. Then $$ -a_{2n}\leq\frac{k_1\cdot H^{1-1/n}}{(\log1/\varepsilon_2)^{1-1/n}}+ \frac{k_2\cdot H^{1+1/n}}{(\log1/\varepsilon_2)^{1/n}}, $$ if $\varepsilon_2=k[\varepsilon(\log T+1)+T^{-1}(\log T)^{1/2n}]$ is sufficient small; $K$ is an absolute constant, $K_1$ and $K_2$ depend on $n$ only.}