随机幂级数在收敛区间端点处的行为研究

Limit Behavior at Convergence Endpoints for Random Power Series

摘要: 本文主要研究实数域中如下形式的随机幂级数$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{a}_n{x}^n}$ ，其中系数$a_n$ 相互独立且仅在有限集$D=\left\{d_1,d_2,\cdots ,d_k\right\}$ 中取值。设$a_n$ 取$d_i$ 的概率为$P_i \left(i=1,2,\cdots ,k\right)$ 且满足${\displaystyle {\sum }_{i=1}^k{P}_i}=1$ 及$P_1\le P_2$ 。我们证明当系数$a_n$ 的期望非零时，幂级数随着$x\to 1^{-}$ 时几乎必然趋向$+\infty$ 或$-\infty$ 。

Abstract: This paper primarily studies random power series of the following form in the real number field$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{a}_n{x}^n}$ , where the coefficients $a_n$ are mutually independent and take values only in a finite set $D=\left\{d_1,d_2,\cdots ,d_k\right\}$ . Suppose the probability that $a_n$ takes the value $d_i$ is $P_i \left(i=1,2,\cdots ,k\right)$ satisfying ${\displaystyle {\sum }_{i=1}^k{P}_i}=1$ and $P_1\le P_2$ . It is proved that when the expectation of the coefficients $a_n$ is non-zero, the power series almost surely tends to $+\infty$ or $-\infty$ as $x\to 1^{-}$ , deterministically depending on the sign of the expectation.