Editing Moment problem (section)

== Probability ==

The moment problem has applications to probability theory. The following is commonly used:<ref>{{Cite web |last=Sodin |first=Sasha |date=March 5, 2019 |title=The classical moment problem |url=https://webspace.maths.qmul.ac.uk/a.sodin/teaching/moment/clmp.pdf |url-status=live |archive-url=https://web.archive.org/web/20220701072907/https://webspace.maths.qmul.ac.uk/a.sodin/teaching/moment/clmp.pdf |archive-date=1 Jul 2022}}</ref>

{{Math theorem|name=Theorem (Fréchet-Shohat)|note=|math_statement= 

If <math display="inline">\mu</math> is a determinate measure (i.e. its moments determine it uniquely), and the measures <math display="inline">\mu_n</math> are such that <math display="block">
      \forall k \geq 0 \quad \lim _{n \rightarrow \infty} m_k\left[\mu_n\right]=m_k[\mu],
      </math> then <math display="inline">\mu_n \rightarrow \mu</math> in distribution.
}}

By checking [[Carleman's condition]], we know that the standard normal distribution is a determinate measure, thus we have the following form of the [[central limit theorem]]:

{{Math theorem
| name = Corollary
| note = 
| math_statement = If a sequence of probability distributions <math display="inline">\nu_n</math> satisfy <math display="block">m_{2k}[\nu_n] \to \frac{(2k)!}{2^k k!}; \quad m_{2k+1}[\nu_n] \to 0</math> then <math display="inline">\nu_n</math> converges to <math display="inline">N(0, 1)</math> in distribution.
}}