# Hamburger Moment Problem with Fractional Moments

It is stated in Lasserre (2010) that a sequence of real numbers $\{y_i\}$ is a valid sequence of integer moments for a positive, finite Borel measure $\mu$ on $\mathbb R$ if and only if the moment matrix defined by $$\mathbf H_n(\mathbf y)= \left [ \begin{array}{cccc} y_0 & y_1 & ... & y_n \\ y_1 & y_2 & ... & y_{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ y_n & y_{n+1} & ... & y_{2n} \end{array} \right ]$$ is positive semidefinite for all $n \in \mathbb N$.

I am wondering: is there a similar necessary and sufficient condition for a sequence of fractional (posynomial) moments $\{ y_{i/3} \}$, where $$y_{i/3}=\int_\mathbb R x^{i/3}d\mu?$$

## 1 Answer

The moments $y_{i/3}$ for random variable $X$ are the integer moments for $X^{1/3}$.

• Thank you for your help, Robert. I agree with your statement. However, I don't see how this gets me anywhere. (Forgive me, I am not a mathematician by training.) Can you say a little more? – Garrett Feb 16 '15 at 23:44
• Here is my best shot: Let $z_i = y_{i/3}$ for all $i \in \mathbb N \cup \{0\}$. If $\mathbf H_n(\mathbf z) \succeq \mathbf 0$ for all $n \in \mathbb N$, then $\{z_i\}$ is a valid sequence of integer moments for some finite positive Borel measure $\mu_Y$, where $Y=X^{1/3}$. I suspect there is then some relationship between the measure $\mu_Y$ and the measure $\mu_X$, corresponding to the random variable $X$, which guarantees that $\mu_X$ is both finite and positive. – Garrett Feb 17 '15 at 0:10
• We may assume we're dealing with probability measures (i.e. $y_0 = 1$). The distribution $\mu_Y$ for $Y = X^{1/3}$ is related to $\mu_X$ for $X$ by $\mu_Y(A) = \mathbb P(Y \in A) = \mathbb P(X \in A^3) = \mu_X(A^3)$ where $A^3 = \{a^3: a \in A\}$. – Robert Israel Feb 17 '15 at 0:36