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.
There exist a probability measure on such that for every
Remarks:
Since is separable point (i.e. convergence in the Lévy–Prokhorov metric) is the same as convergence in distribution . If we remove the symmetric distribution condition:
in a finite-dimensional setting equivalence is true for all except point (i.e. the uniform tighness of ),[2]
in an infinite-dimensional setting is true but does not always hold.[2]: 37
Literature
Pap, Gyula; Heyer, Herbert (2010). Structural Aspects in the Theory of Probability. Singapore: World Scientific. p. 79.