Glivenko's theorem (probability theory)

In probability theory, Glivenko's theorem states that if ${\displaystyle \varphi _{n},n\in \mathbb {N} }$, ${\displaystyle \varphi }$ are the characteristic functions of some probability distributions ${\displaystyle \mu _{n},\mu }$ respectively and ${\displaystyle \varphi _{n}\to \varphi }$ almost everywhere, then ${\displaystyle \mu _{n}\to \mu }$ in the sense of probability distributions.[1]

References

1. Itô, Kiyosi (1984). Introduction to Probability Theory. Cambridge University Press. p. 87. ISBN 0 521 26960 1.