Remarks on lower estimates for characteristic functions of probability laws

Анотація

For the analytic in 𝔻_R={z:|z|<R} characteristic function ϕ of a probability law F it is investigated conditions on W_F(x)=1-F(x)+F(-x) (x≥0) and a positive continuous function h increasing to +∞, under which h(ln(M(r,ϕ)))≥(1+o(1))/(R-r) or ln(M(r,ϕ))≥(1+o(1))h(1/(R-r)) as r↑R, where M(r,ϕ)=max{|ϕ(z)|:|z|=r<R}.