On the distribution of multiple power series regularly varying at the boundary point

Let $B(x)$ be a multiple power series with nonnegative coefficients which is convergent for all $x\in(0,1)^n$ and diverges at the point $\mathbf1=(1,\dots,1)$. Random vectors (r.v.) $\xi_x$ such that $\xi_x$ has distribution of the power series $B(x)$ type is studied. The integral limit theorem for r.v. $\xi_x$ as $x\uparrow\mathbf1$ is proved under the assumption that $B(x)$ is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series $B(x)$ are one-sided weakly oscillating at infinity.