The stability of solutions of the functional equations connected with characterization theorems for probability distributions

N. A. Sapogov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 15-25

Résumé

The present paper contains some results on the solutions $\Psi_j(x)$ of the functional inequality \begin{equation} \biggl|\sum\Psi_j(a^T_jt)\biggr|\leq\varepsilon, \tag{1} \end{equation} where $a^T_j=(a_{1j},a_{2j},\dots,a_{pj})\in\mathbb{R}^p$ all the coefficients $a_{ij}$ are constants, $t=(t_1,t_2,\dots,t_p)\in\mathbb{R}^p$, $a_j^Tt=\sum_{i=1}^p a_{ij}t_i$, $p\geq2$, the relation (1) holds for all $t_j\in\mathbb{R}^1$, $j=1,2,\dots,n$. Inequality (1) is connected with certain characterization theorem in theory of probability and statistics. For the sake of simplicity we suppose that $\Psi_j(x)$ are continuous functions, $x\in\mathbb{R}^1$. We obtain the following main results Theorem. {\it Let (1) holds, $n\ge 1$, $p=2$, $\Delta_{kj}=a_{1j}a_{2k}-a_{1k}a_{2j}\neq0$ for $j\ne k$, $\varepsilon>0$ is an arbitrary positive number. Then there exist polynomials $P_{n,j}$, $j=1,\dots,n$, such that $$ \biggl|\Psi_j(x)-P_{n,j}(x)\biggr|\le 4^{n-2}\varepsilon $$ for all $x\in\mathbb{R}^1$, $j=1,2,\dots,n$. The degrees of $P_{n,j}(x)$ are $\leq n-2$.} The particular case $n=3$, $p=2$ is of some interest and was investigated in more details.