Itô–Nisio_theorem
Itô–Nisio theorem
Convergence of random variables in Banach spaces
The Itô-Nisio theorem is a theorem from probability theory that characterizes convergence in Banach spaces. The theorem shows the equivalence of the different types of convergence for sums of independent and symmetric random variables in Banach spaces. The Itô-Nisio theorem leads to a generalization of Wiener's construction of the Brownian motion.[1] The symmetry of the distribution in the theorem is needed in infinite spaces.
The theorem was proven by Japanese mathematicians Kiyoshi Itô and Makiko Nisio in 1968.[2]