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$\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\delta y$, where $M(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ is the generating function for a certain probability distribution $\{s_{n,k}\}$ (the exact formula for $s_{n,k}$ is unknown), and $\delta$, $\beta$, $\mu$ are all constants.

The problem comes out of a probability model, and data show that the distribution should have a finite mean and infinite second moment ($M_{xx}(1,1)=\infty$). My question is that is there any way (or any theory on it) to get the asymptotic of $M_{xx}(x,1)$ when $x\rightarrow 1^{-}$? (by asymptotic I mean something like $M_{xx}(x,1)\sim C(1-x)^{-\zeta}$)

The background for this problem is here: Asymptotic behaviour of a two-dimensional recurrence relation

Thank you!

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    Thank you for your reply. But in my model, though $\delta$ should be small (it is supposed to be the rate of exit of firms), it has to be a positive number. And $\beta$ (supposed to be job-filling rate) should be very large.2011-08-15

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