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This problem comes out of a research in models of firm growth.

The model is simple: A firm has two parameters which are its size (number of employees) and job vacancies. A firm of size $n$ will produce job vacancies at rate $n\mu$ and $k$ job vacancies will be filled at rate $k\beta$. On the other hand, a firm is likely to quit at rate $\delta$, and new firms come into the market at rate $\alpha$ (new firm starts with 0 employee and 1 vacancy, but one can also model it as 1 employee and 0 vacancy if necessary).

So in terms of math, the equation describing the relation is: $D_t s_{n,k,t}=-(n\mu+k\beta+\delta)s_{n,k,t}+(k+1)\beta s_{n-1,k+1,t}+n\mu s_{n,k-1,t}$, when (n,k) is not in the boundary, where $s_{n,k,t}$ is the probability of a firm having size $n$ and $k$ vacancies.

I am particularly interested in the stationary distribution which is described by:
$s_{0,1}=\frac{\alpha}{\delta+\beta}$,
$s_{n,k}=0$ for $n<0$ or $k<0$,
and $(n\mu+k\beta+\delta)s_{n,k}=(k+1)\beta s_{n-1,k+1}+n\mu s_{n,k-1}$.

Moreover, the generating function $M(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ satisfies the PDE $\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\alpha y$.

What I would like to know is the asymptotic behavior of $s_n=\sum_{k=0}^{\infty}{s_{n,k}}$, or more explicitly the tail $\sum_{n=m}^{\infty}{s_n}$ when m is very large. The empirical Data on firm size suggests that it might appear as a power law (~ $n^{-\zeta}$, where $\zeta$ is a bit larger than 1), also it may not have a finite variance ($M_{xx}(1,1)=\infty$). The one-dimensional problem, in which we only consider the size (that is, vacancies are filled immediately), is completely characterized by Yule process and also suggests the power law.

Could anyone give me some insight on how to approach this problem? I've been considering this problem for a month, but since I am not familiar with highly technical tools in analysis or probability I get stuck completely.

Thank you!

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    What I was mainly thinking about was to find out the behaviour of M_xx(x,1) when x->1 (because M_xx doesn't have a finite variance), and then use some versions of tauberian theorem to get the tail. It actually worked for a modified model where the PDE for M was soluble. (I used results from this paper: http://epubs.siam.org/sidma/resource/1/sjdmec/v3/i2/p216_s1). But that model was kind of "artificial".2011-08-12

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