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Let $μ=0,σ>0$. What is the maximum entropy distribution over the integers with mean $μ$ and variance $σ^2$?

Is Skellam distribution a maximum entropy distribution? Is there a closed-form expression to compute the entropy of this distribution?

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    $\begingroup$ The wikipedia article en.wikipedia.org/wiki/Maximum_entropy_probability_distribution already tells you that it must be of the form $p_n= c e^{an-bn^2}$, and it would seem $a,b,c$ are determined by your requirements on $\mu,\sigma$. $\endgroup$
    – Christian Remling
    Commented Nov 21, 2017 at 16:53
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    $\begingroup$ Noting that $\sum_{n=-\infty}^\infty \exp(-bn^2) = \theta_3(0,\exp(-b))$ (a Jacobi Theta function), you want $a=0$, $c = 1/\theta_3(0,\exp(-b))$, and $$\dfrac{d}{db} \theta_3(0, \exp(-b)) = -\sigma^2 \theta_3(0,\exp(-b))$$ $\endgroup$
    – Robert Israel
    Commented Nov 21, 2017 at 19:20
  • $\begingroup$ Could you refer to any book or article on those results? Many thanks! $\endgroup$
    – Ioannis Papoutsidakis
    Commented Nov 21, 2017 at 20:52
  • $\begingroup$ Wikipedia and links there. $\endgroup$
    – Robert Israel
    Commented Nov 22, 2017 at 21:02

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