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I have a probability generating function

$$ G(z) = \Bigg(\frac{ 1-2d + \sqrt{1-4d(1-d)z}}{2(1-2d)}\Bigg)^{\frac{1-2d}{d^2}\kappa}\ \Bigg(\frac{1-\sqrt{1-4d(1-d)z}}{2dz}\Bigg)^{\frac{\kappa}{d^2}}$$

and I'd like to expand this for $d\rightarrow 0$. This expansion should retain the property $G(1)=1$.

My attempt was to use $$ \sqrt{1-4d(1-d)z} \approx 1-2dz$$ within both numerators of this expression, and this obtains $$G(z) \approx \Bigg(\frac{1-\frac{d}{1-d}z}{1-\frac{d}{1-d}}\Bigg)^{\frac{1-2d}{d^2}\kappa}. $$

However, I do not know if this is valid, since I neglected terms of $O(d^2)$ in the two terms, while I kept terms of $O(1/d^2$) in the exponent...

Is there some better way to do this limit $d\rightarrow 0$?

kevinkayaks
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    $G(z)=e^{\kappa (z-1)} \left(1+d \kappa \left(z^2-1\right)\right)+\mathcal O(d^2)$. – AccidentalFourierTransform Oct 03 '19 at 02:11
  • I believe that, since this limit shows a Poisson pgf as $d\rightarrow 0$, but how do you get the exponential factor? I was also hoping to show a transition from its full form to NegBin and progressively Poisson as $d$ gets smaller. Not sure if that's buried in the structure of $G$. – kevinkayaks Oct 03 '19 at 03:53

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