On the belonging of analytic in unit disc characteristic functions of probability laws to generalized convergence class

Анотація

For  continuous on $[x_0,+\infty )$ functions $\alpha$ and $\beta$ increasing to $+\infty$ we say that an analytic in $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$ characteristic function  $\phi$ of a probability law $F$ belongs to the  generalized convergence $\alpha \beta$-class if ${\int }_{r_0}^1\frac{\alpha \left(\ln M(r,\phi )\right)}{(1-r{)}^2\beta \left(\frac{1}{1-r}\right)}dr<+\infty$, where  $M\left(r,\phi \right)=max\left\{\phi \left(z\right): \left|z\right|=r\right\}$. Conditions on $\alpha$, $\beta$ and $F$ are found under which the function $\phi$ belongs to the generalized convergence $\alpha \beta$-class if and only if ${\int }_{x_0}^{\infty }\alpha \text{'}\left(x\right){\beta }_1\left(\frac{x}{\ln \left(W_F\left(x\right)e^x\right)}\right)dx<+\infty$, where ${\beta }_1\left(x\right)={\int }_x^{\infty }\frac{dt}{\beta \left(t\right)}$ and $W_F\left(x\right)=1-F\left(x\right)+F(-x)$.