The following theorem is from "A property of power series with positive coefficients" by P. Erdos et al.
Let $p_k$ be a sequence of nonnegative numbers for which $\sum_{k=0}^{\infty}p_k=1$ and let $m=\sum_{k=0}^{\infty}kp_k\le\infty$. If $P(x)=\sum_{k=0}^{\infty}p_kx^{k}$ is not a power series in $x^t$ for any integer $t>1$, then the series $$ U(x)=\frac{1}{1-P(x)}=\sum_{k=0}^{\infty}u_kx^{k} $$ has the property $\lim_{n\to\infty}u_n=1/m$.
Actually, the theorem could be used to study the limit of a sequence $u_n$ with $u_0=1$, $u_n+\sum_{j=0}^{n-1}u_jp_{n-j}=1$.
I was wondering if there is a similar result that could be used to study the limit behaviour if $u_n$ satisfies the following condition:
$u_0=1$, $b_{k,0}=1$ for each $k\ge0$ and $\sum_{j=0}^{k}u_{j}b_{j,k-j}=1$ where each $0<b_{j,k-j}<1$. Any hint, reference or suggestion would be of great help. Thank you very much.