Asymptotic behaviour of a recursively defined sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:06:24Z http://mathoverflow.net/feeds/question/71355 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71355/asymptotic-behaviour-of-a-recursively-defined-sequence Asymptotic behaviour of a recursively defined sequence Epsilon 2011-07-26T21:09:50Z 2011-07-26T21:17:26Z <p>I encounter a problem in which I need to characterize the asymptotic behaviour of a sequence.</p> <p>${s_{n,k}}$ is a stationary probability distribution, i.e., $\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}}=1}$, and satisfies <br/> $s_{0,1}=\frac{\alpha}{\delta+\beta}$, <br/> $s_{n,k}=0$ for $n&lt;0$ or $k&lt;0$, <br/> and $(n\mu+k\beta+\delta)s_{n,k}=(k+1)\beta s_{n-1,k+1}+n\mu s_{n,k-1}$, <br/> where $\alpha, \delta, \beta, \mu$ are constants.</p> <p>What I want to know is the asymptotic behaviour of $\sum_{n=m}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}}}$ when $m \rightarrow \infty$</p> <p>Initially I considered the generating function $F(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ which satisfies the PDE $\delta F=\beta(x-y)F_y+\mu x(y-1)F_x+\alpha y$, but I didn't find anything useful since this PDE doesn't seem solvable.</p> <p>Can anyone suggest a way to do it or suggest references? This problem arises in an economic model for firms (you can think of $n$ as size and $k$ as the number of vacancies, and thus the recursive relation describes how firm size grows). It's like a generalization of the birth-death process, but is more complicated.</p> <p>Thank you for your help!</p>