$$ \sum_i,j=0^N a_i a_j m_i+j \ge 0 $$
encodes all the moments. The measure is determinate iff the associated (a tridiagonal matrix) is essentially self-adjoint in $\ell^2$. Indeterminacy corresponds to a deficiency of self-adjoint extensions—a concept from quantum mechanics. Complex Analysis and the Stieltjes Transform Define the Stieltjes transform of $\mu$: $$ \sum_i,j=0^N a_i a_j m_i+j \ge 0 $$
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$ Complex Analysis and the Stieltjes Transform Define the
for all finite sequences $(a_0,\dots,a_N)$. This means the infinite $H = (m_i+j)_i,j=0^\infty$ must be positive semidefinite (all its finite leading principal minors are $\ge 0$). In probability and analysis, a moment is a
At first glance, this seems like a straightforward problem of "matching moments." But as we will see, it opens a Pandora's box of deep analysis, touching functional analysis, orthogonal polynomials, complex analysis, and even quantum mechanics. In probability and analysis, a moment is a generalization of the idea of "average power." For a real random variable $X$ with distribution $\mu$ (a positive measure on $\mathbbR$), the $n$-th moment is: