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Fix $\mathbf t \in \mathbb{R}_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}_{+}^{d}$ that solves the following 'moment' conditions.

$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, \lvert\mathbf i \rvert \le n \in \mathbb N, \; \mu_{\mathbf i}^{\mathbf{t}} = \int\limits_{\mathbb R_+^d} \frac{\prod\limits_{j=1}^d r_j^{i_j}}{\left(1+\sum\limits_{j=1}^{d} r_j t_j \right)^{\lvert \mathbf i \rvert}}\nu\left(d\mathbf r\right)$$

For given values of $\mu_{i}^{t}$.

Is there a transformation of my equations (by a push-forward change on $\nu$ for exemple) that would translate to a more 'classical' multivariate truncated moment problem ? i.e, i want an expression without the bottom part of the fraction..

Edit : My idea is currently to increase the dimension by one, by the change of variable :

$$\mathbf p = \left(r_1,...,r_d,\frac{1}{\left(1+\sum\limits_{j=1}^{d} r_j t_j \right)}\right)$$

Since $\mathbf t$ is fixed, this change of variable might be bijective, and it makes $\mu_{\mathbf{i}}^{\mathbf t}$ the $\left(i_1,...,i_d,\lvert\mathbf i \rvert\right)$th moment of an other random variable.. Will this work ? I am not very confident about which algorithm exists to solve such a r-atomic moment problem.

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Changing the variables from $\mathbf r=(r_j)$ to $\mathbf s=(s_j)$, where $$s_j:=g(\mathbf r)_j:=\frac{r_j}{1+\sum_i t_ir_i},$$ we have $$r_j:=\frac{s_j}{1-\sum_i t_is_i}.$$ So, the transformation $g$ is a homeomorphism of $\mathbb R_+^d$ onto $$\Sigma:=\Big\{\mathbf s\in\mathbb R_+^d\colon \sum_i t_is_i<1\Big\}.$$ Therefore, the problem can indeed be rewritten as the multivariate moment problem to find a measure $\rho$ such that $$\mu_{\mathbf i}^{\mathbf{t}}=\int_\Sigma \rho(d\mathbf s)\, \prod_{j=1}^d s_j^{i_j}\quad \forall\, \mathbf i \in \mathbb{N}^{d} \ s.t. \ \lvert\mathbf i \rvert \le n \in \mathbb N. $$ The measure $\nu$ can then be obtained by the formula $\nu(B)=\rho(g(B))$ for all Borel $B\subseteq\mathbb R_+^d$.

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  • $\begingroup$ Waouh. Thanks, i was sure that it could be done but i did not found the $g_j$ functions. You made my day. I now have to found how the moment problem can be solved numericaly, but this is another story. $\endgroup$
    – lrnv
    Jun 8, 2020 at 17:45
  • $\begingroup$ I am glad this helped. $\endgroup$ Jun 8, 2020 at 17:56

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