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

In the paper we consider entire functions $fcolon mathbb{C}^ptomathbb{C}, pgeq 2,$ 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}}, r^{wedge}=min{r_icolon iin{1,ldots,p}}$ and let $l$ be a log-convex real function on $(1,+infty)$ such that $ln t=o(l(t)), tto+infty.$ Then for any entire transcendental function $f$ {with} $ln M_f(r)leq l(r^{vee}), r^{wedge}to+infty,$ {the} inequality $varlimsuplimits_{r^{wedge}to+infty} frac{ln M_f(r)-lnmu_f(r)}{lnlnmu_f(r)}leqalpha$ holds if and only if $ varlimsuplimits_{tto+infty}(ln l(t)/lnln t)leq1+alpha/p.$ Similar theorems are proved for random entire functions of several complex variables.