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](https://en.wikipedia.org/wiki/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!



Exponentially Bounded Sequence of Moments defining Distribution?
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. 


