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I have an exponentially bounded sequence $m_n = \lambda^n + c_n$ (i.e. the $c_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying the Hamburger Moment Problem, which means I would have to show that the Hankel kernel of the matrix

$$A = \left(\begin{array}{ccc} m_{0} & m_{1} & \ldots\\ m_{1} & m_{2} & \ldots\\ \vdots & \vdots & \ddots \end{array}\right)$$

is positive definite. Is it known that this is true for such a sequence?

Thanks in advance!

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  • $\begingroup$ If the $c_n$ are exactly quadratic in n, say $a(n)(n-1) + bn + c$, then the moment generating function can be calculated. $\endgroup$
    – Michael
    Commented Nov 6, 2015 at 15:41

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No. Because $c_n$ is quadratic, the values of $m_0, m_1, m_2$ can be arbitrary (subject to $m_0 = 1$ since presumably we are looking at a probability distribution). The first condition that needs to be satisfied is that $m_0 m_2 - m_1^2 \ge 0$, and by tweaking $c_n$ appropriately you can easily arrange for this to be false.

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