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 \begin{equation} \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 ] \end{equation} 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 \begin{equation} y_{i/3}=\int_\mathbb R x^{i/3}d\mu? \end{equation}


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

  • $\begingroup$ 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? $\endgroup$ – Garrett Feb 16 '15 at 23:44
  • $\begingroup$ 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. $\endgroup$ – Garrett Feb 17 '15 at 0:10
  • $\begingroup$ 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 $A^3 = \{a^3: a \in A\}$. $\endgroup$ – Robert Israel Feb 17 '15 at 0:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.