Itô–Nisio theorem explained

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 in 1968.^[2]

Statement

Let

(E,\| ⋅ \|)

be a real separable Banach space with the norm induced topology, we use the Borel σ-algebra and denote the dual space as

E^*

. Let

\langle

z,S\rangle:={}
	E^*

\langlez,S\rangle_E

be the dual pairing and

is the imaginary unit. Let

X_1,...,X_n

be independent and symmetric

-valued random variables defined on the same probability space

S_n=\sum

	n

	i=1

X_n

\mu_n

be the probability measure of

S_n

some

-valued random variable.The following is equivalent^[2]

S_n\toS

converges almost surely.

S_n\toS

converges in probability.

\mu_n

converges to

\mu

in the Lévy–Prokhorov metric.

\{\mu_n\}

is uniformly tight.

\langlez,S_n\rangle\to\langlez,S\rangle

in probability for every

z\inE^*

There exist a probability measure

\mu

such that for every

z\inE^*

	i\langlez,S_n\rangle
E[e

]\to\int_Ee^i\langle\mu(dx).

Remarks:Since

is separable point

(i.e. convergence in the Lévy–Prokhorov metric) is the same as convergence in distribution

\mu_n\implies\mu

. 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

\{\mu_n\}

in an infinite-dimensional setting

1\iff2\iff3

is true but

6\implies3

does not always hold.

Literature

Book: Gyula. Pap. Herbert. Heyer. Structural Aspects in the Theory of Probability. Singapore. World Scientific. 2010. 79.

References

Nobuyuki. Ikeda. Setsuo. Taniguchi. 2010. 10.1016/j.spa.2010.01.009. 5. 605–621. Stochastic Processes and Their Applications. The Itô–Nisio theorem, quadratic Wiener functionals, and 1-solitons. 120. free.
Kiyoshi. Itô. Makiko. Nisio . Osaka University and Osaka Metropolitan University, Departments of Mathematics . On the convergence of sums of independent Banach space valued random variables . Osaka Journal of Mathematics . 5 . 1 . 1968 . 35–48 .