The central question of the is: Can you uniquely reconstruct the contents of the box—specifically, a measure or a probability distribution—from this infinite sequence of moments?
Two powerful criteria from analysis:
: Always determinate. The moments uniquely define the measure due to the Weierstrass Approximation Theorem Hamburger/Stieltjes Problems : Can be indeterminate. For example, the log-normal distribution The central question of the is: Can you
For a sequence to be a moment sequence, it must satisfy specific positivity conditions. Generally, this is characterized by the positive definiteness of —matrices formed by the sequence elements where For example, the log-normal distribution For a sequence
We assume all moments exist (are finite). The classical moment problem asks: Given a sequence $(m_n)_n=0^\infty$, does there exist some measure $\mu$ that has these moments? If yes, is that measure unique? If yes, is that measure unique
In quantum mechanics, moments of position correspond to expectation values $\langle x^n \rangle$. The question "Is the Hamiltonian self-adjoint?" is intimately related to the moment problem. The classic example: the "Stieltjes moment problem" appears in the study of the anharmonic oscillator $H = p^2 + x^4$. The measure of the ground state is determinate, guaranteeing a unique quantum theory.