Multivariate truncated moments problems and maximum entropy

Abstract

We characterize the existence of the Lebesgue integrable solutions of the truncated problem of moments in several variables on unbounded supports by the existence of some maximum entropy—type representing densities and discuss a few topics on their approximation in a particular case, of two variables and 4th order moments.

Appendix: The functions \({r}_{{j\, i}}\)

We give an algorithm to recurrently compute $r_{ji},c_{ji}$, in particular solve (12) to finish the proof of Proposition 8. Set $\delta _k =\delta _{(l+1-k,k)\, j_0 }$ for $0\le k\le l+1$. Let $\alpha _k =a_{(l-2-k,k)}$, $\beta _k =b_{(l-2-k,k)}$ for $0\! \le \! k\! \le \! l\! -\! 2$. Thus $\alpha _k ,\beta _k =0$ for $k<0$, $k\ge l-1$. Also $x_\kappa =0$ if $\kappa \not \ge 0$. Change the summation indices in (12) by $j\! =\! i\! +\! e_{1,2}\! -\! \iota$ ($\ge \! 0$). Then (12) becomes $\sum _{|j|=l-2, (j_2 \le i_2) }(i_1 -j_1 +1)x_{i-j+e_1}b_j -\sum _{|j|=l-2,( j_2 \le i_2)}(i_2 -j_2 +1)x_{i-j+e_2}a_j =\delta _{ij_0}$ where the (redundant) condition $j_2\le i_2$ follows from $j\le i$, that comes from $\iota \ge e_{1,2}$. For every $i=(l+1-k,k)$ with $0\le k\le l+1$, we have the equivalence $(j\ge 0,|j|=l-2, j_2 \le i_2)\, \Leftrightarrow \, j=(l-2-p,p)$ for $0\le p\le k$ and hence the $l+1$ equations in (12) become now, respectively,

$$\begin{aligned} \sum _{p=0}^k [ (4\! +\! p\! -\! k)x_{(4+p-k,k-p)}\beta _p \! - \! (k\! -\! p\! +\! 1)x_{(3+p-k,k-p+1)}\alpha _p ] \! =\! \delta _k ,\quad 0\! \le \! k\! \le \! l\!+\!1.\nonumber \\ \end{aligned}$$

(15)

If $l\ge 5$, let $\alpha _0 ,\ldots ,\alpha _{l-5}=0$ and define $\beta _0 ,\ldots ,\beta _{l-5}$ inductively by$4x_{40}\beta _k = -\sum _{p=0}^{k-1}(4+p-k)x_{(4+p-k,\, k-p)}\beta _p +\delta _k$ ($0\le k\le l-5$) where $\sum _\emptyset :=0$. Note that $x_{40}<0$ since $g\in G$. This fulfills (15) for $0\le k\le l-5$. Last six equations in (12) [($l-4\le k\le l+1$ in (15)] will provide $\alpha _k ,\beta _k$ ($l-4\le k\le l-2$). If $l=4$, skip this step and go directly to the linear $6\times 6$ system [(in this case (16) for $y,z,w=0$]. In any case, we let now $i=(l+1-k,k)$ for $0\le k\le l+1$ in (12). We have $i+e_{1}-\iota =(l+2-k -\iota _1 ,k-\iota _2)$ and $i+e_2 -\iota =(l+1-k-\iota _1 ,k-\iota _2 +1)$. Last 6 equations in (12) become $\sum _{|\iota |=4} \! \iota _1 x_\iota \beta _{k -\iota _2 } \! -\! \sum _{|\iota |=4} \! \iota _2 x_\iota \alpha _{k-\iota _2 +1}=\delta _k$ ($l-4\le k\le l+1$), see below. The brackets $( )$ border quantities already known in terms of $\beta _0 ,\ldots ,\beta _{l-5}$. The markers $\lfloor \rceil$ border sums of terms that are null due to $\iota _{1,2}=0$, $\alpha _k ,\beta _k =0$ ($k\ge l-1$) or $\alpha _k =0$ ($0\le k\le l-5$):

$$\begin{aligned} \begin{array}{l} 4x_{40}\beta _{l-4}+(3x_{31}\beta _{l-5}+2x_{22}\beta _{l-6}+1x_{13}\beta _{l-7}+0x_{04}\beta _{l-8})\\ \lfloor -0x_{40}\alpha _{l-3}\rceil -1x_{31}\alpha _{l-4}\, \lfloor -2x_{22}\alpha _{l-5}-3x_{13}\alpha _{l-6}-4x_{04}\alpha _{l-7}\, \rceil =\delta _{l-4} \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l}4x_{40}\beta _{l-3}+3x_{31}\beta _{l-4}+(2x_{22}\beta _{l-5}+1x_{13}\beta _{l-6}+0x_{04}\beta _{l-7})\\ \lfloor -0x_{40}\alpha _{l-2}\rceil -1x_{31}\alpha _{l-3}-2x_{22}\alpha _{l-4}\, \lfloor -3x_{13}\alpha _{l-5}-4x_{04}\alpha _{l-6}\, \rceil =\delta _{l-3} \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l}4x_{40}\beta _{l-2}+3x_{31}\beta _{l-3}+2x_{22}\beta _{l-4}+(1x_{13}\beta _{l-5}+0x_{04}\beta _{l-6})\\ \, \lfloor -0x_{40}\alpha _{l-1}\, \rceil -1x_{31}\alpha _{l-2}-2x_{22}\alpha _{l-3}-3x_{13}\alpha _{l-4}\, \lfloor -4x_{04}\alpha _{l-5}\, \rceil =\delta _{l-2} \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l}\, \lfloor 4x_{40}\beta _{l-1}+\, \rceil \, 3x_{31}\beta _{l-2}+2x_{22}\beta _{l-3}+1x_{13}\beta _{l-4}+\lfloor 0x_{04}\beta _{l-5}\rceil \\ \, \lfloor -0x_{40}\alpha _{l}-1x_{31}\alpha _{l-1}\, \rceil -2x_{22}\alpha _{l-2}-3x_{13}\alpha _{l-3}-4x_{04}\alpha _{l-4} =\delta _{l-1}\end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l}\, \lfloor 4x_{40}\beta _{l}+3x_{31}\beta _{l-1}+\, \rceil \, 2x_{22}\beta _{l-2}+1x_{13}\beta _{l-3}+\lfloor 0x_{04}\beta _{l-4}\rceil \\ \, \lfloor -0x_{40}\alpha _{l+1}-1x_{31}\alpha _{l}-2x_{22}\alpha _{l-1}\, \rceil -3x_{13}\alpha _{l-2}-4x_{04}\alpha _{l-3} =\delta _{l} \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l}\, \lfloor 4x_{40}\beta _{l+1}+3x_{31}\beta _{l}+2x_{22}\beta _{l-1}+\, \rceil \, 1x_{13}\beta _{l-2}+\lfloor 0x_{04}\beta _{l-3}\rceil \\ \, \lfloor -0x_{40}\alpha _{l+2}-1x_{31}\alpha _{l+1}-2x_{22}\alpha _{l}-3x_{13}\alpha _{l-1}\, \rceil -4x_{04}\alpha _{l-2} =\delta _{l+1}. \end{array} \end{aligned}$$

Set $y\! =\! -3x_{31}\beta _{l-5} \! -\! 2x_{22}\beta _{l-6}\! -\! x_{13}\beta _{l-7}$, $z\! =\! -2x_{22}\beta _{l-5}\! -\! x_{13}\beta _{l-6}$ and $w\! =\! -x_{13}\beta _{l-5}$.

We easily read from above that $\alpha _k ,\beta _k$ for $k\! =\! l-4,\, l-3,\, l-2$ are given by

$$\begin{aligned} \left[ \, \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 4x_{40}&0&0&x_{31}&0&0\\ [2mm] 3x_{31}&4x_{40}&0&2x_{22}&x_{31}&0\\ [2mm] 2x_{22}&3x_{31}&4x_{40}&3x_{13}&2x_{22}&x_{31}\\&\,&\,&\,&\,&\\ x_{13}&2x_{22}&3x_{31}&4x_{04}&3x_{13}&2x_{22}\\ [2mm] 0&x_{13}&2x_{22}&0&4x_{04}&3x_{13}\\ [2mm] 0&0&x_{13}&0&0&4x_{04} \end{array}\, \right] \! \left[ \, \begin{array}{l}\beta _{l-4}\\ [2mm] \beta _{l-3}\\ [2mm] \beta _{l-2}\\ \\ -\alpha _{l-4}\\ [2mm] -\alpha _{l-3}\\ [2mm] -\alpha _{l-2} \end{array} \, \right] \! \! =\! \! \left[ \begin{array}{l}y+\delta _{l-4}\\ [2mm] z+ \delta _{l-3}\\ [2mm] w+\delta _{l-2}\\ \\ \delta _{l-1}\\ [2mm] \delta _{l}\\ [2mm] \delta _{l+1} \end{array}\right] \end{aligned}$$

(16)

(note also that $g\in G_0$). We have $a_j, b_j$, and so $u, v$ such that $\text{ deg}\, (L(u,v)-t^{j_0})\le l$. Now $\pi =t^{j_0}-L$ is determined by summing the terms of degree $\le l$ in $-L$. For $m=1,2$ set $K_m = \{ (j,\iota ): |j|=l-2, |\iota |\le 3, \iota _m \ge 1\}$. Then

$$\begin{aligned} \pi&= \sum _{(j,\, \iota )\in K_2 }\! a_j \iota _2 x_\iota t^{j+\iota -e_2}\! -\! \sum _{(j,\, \iota )\in K_1 }\! b_j \iota _1 x_\iota t^{j+\iota -e_1}\! +\sum _{|j|=l-2, j_2 \ge 1}\! j_2 a_j t^{j-e_2}\\&- \sum _{|j|=l-2, j_1 \ge 1}\! j_1 b_j t^{j-e_1}. \end{aligned}$$

For any $i\ge 0$ with $|i|\le l$, the coefficient of $t^i$ in the sum $\Sigma _{K_2}$ from above is $\sum _{(j,\, \iota )\in K_2 (i) }a_j \iota _2 x_\iota$ where $K_2 (i)= \{ (j,\iota )\in K_2 :j+\iota -e_2 =i\}$. The map $K_2 (i)\ni (j,\iota )\mapsto i-j$ is bijective onto $I_i :=\{ \kappa \ge 0 :\kappa \le i,|\kappa |=|i|+2-l\}$. Then we may use it to change the summation index by $\kappa =i-j$ and get the coefficient of $t^i$ in $\Sigma _{K_2}$ as $\sum _{\kappa \in I_i} (\kappa _2 +1)a_{i-\kappa }x_{\kappa +e_2}$. Similarly, the coefficient of $t^i$ in $\Sigma _{K_1}$ is $\sum _{\kappa \in I_i}(\kappa _1 +1)b_{i-\kappa }x_{\kappa +e_1}$. The coefficient $c_{j_0 i}$ ($=\,$a rational function $c_{\, lj_0 i}(x)$ of $x$, actually) of $t^i$ in $\pi (t)$ is then

$$\begin{aligned} c_{j_0 i} =\sum _{\kappa \in I_i}[(\kappa _2 +1)x_{\kappa +e_2}a_{i-\kappa } -(\kappa _1 +1)x_{\kappa +e_1}b_{i-\kappa } ]+d_{j_0 i}\quad (|i|\le l) \end{aligned}$$

(17)

where $d_{j_0 i}\! =\! (i_2 +1)a_{i+e_2 }\! -\! (i_1 +1)b_{i+e_1 }$ if $|i|\! =\! l-3$, and 0 otherwise. We have

$$\begin{aligned} g_{j_0} =\sum _{|i|\le l}c_{j_0 i}(x)\, g_i \text{(} |j_0 |=l+1,\, l\ge 4\text{)}.\end{aligned}$$

(18)

Successive compositions of the mapping $(g_i)_{|i|\le l}\mapsto ((g_{j_0})_{|j_0 |=l+1},\, (g_i)_{|i|\le l})=$ $(g_i)_{|i|\le l+1}$ given by (18) for $l=4,5,\ldots$ provide us with $r_{ji}(x)$ such that

$$\begin{aligned} g_{j}=\sum _{|i|\le 4}r_{ji}(x)g_i \text{(} |j|\ge 5\text{)}. \end{aligned}$$

(19)

Thus (16)–(19) provide $c_{ji},r_{ji}$. Since $\det A x \! \not =\! 0$ and $x_{40}\! =\! \sum _{|i|=4}x_{i}t^i |_{t=e_1 }\! <\! 0$, the denominators of the rational functions $r_{ji}$ do not vanish at $x=\lambda ^*$. $\square$

It would be interesting to generalize Proposition 8 to arbitrary $n$ and $k$, for a class of simple domains $T$ including $\mathbb R ^n$, $[0,\infty )^n$ and get rid of assumptions like $g\in G_0 , G$, for Lagrangians $L_\epsilon$ with $\epsilon >0$. Also, numerical tests of systems like (13) could be tried.