Wiman’s type inequalities without exceptional sets for random entire functions of several variables

In the paper we {consider entire} functions $ fcolonmathbb{C}^p omathbb{C}, pgeq 2, $ defined by power series$ f(z)=f(z_1,ldots,z_p)=sum_{|n|=0}^{+infty}a_n z^n, %pgeq2, $ $z^n=z_1^{n_1}cdotldotscdot z_p^{n_p},$$n=(n_1,ldots,n_p).$ For $r=(r_1,ldots,r_p)inmathbb{R}_+^p$ we {set} $M_f(r)=max{|f(z)|colon|z_i|leq r_i, iin{1,ldots,p}}, mu_f(r)=max{|a_n|r^{n}colon ninmathbb{Z}_+^p},$$ r^{vee}=max{r_icolon iin{1,ldots,p}}, ^{wedge}=min{r_icolon iin{1,ldots,p}}$ and {let $l$be a} log-convex real function on $(1,+infty)$ such that $lnt=o(l(t)), t o+infty.$ Then for any entire transcendentalfunction $f$ {with} $ln M_f(r)leq l(r^{vee}), ^{wedge} o+infty,$ {the} inequality$varlimsuplimits_{r^{wedge} o+infty} frac{lnM_f(r)-lnmu_f(r)}{lnlnmu_f(r)}leqalpha$ holdsif and only if $ varlimsuplimits_{t o+infty}(lnl(t)/lnln t)leq1+alpha/p. $ Similar theorems are proved for random entire functions of several complex variables.