Itô–Nisio theorem

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Let ${\displaystyle (E,\|\cdot \|)}$ be a real separable Banach space with the norm induced topology, we use the Borel σ-algebra and denote the dual space as ${\displaystyle E^{*}}$. Let ${\displaystyle \langle z,S\rangle$ :={}_{E^{*}}\langle z,S\rangle _{E}} be the dual pairing and ${\displaystyle i}$ is the imaginary unit. Let

• ${\displaystyle X_{1},X_{2},\ldots }$ be an infinite sequence of independent and symmetric ${\displaystyle E}$-valued random variables defined on the same probability space,
• ${\displaystyle S_{n}=\sum _{k=1}^{n}X_{k}}$,
• ${\displaystyle \mu _{n}}$ be the probability measure of ${\displaystyle S_{n}}$,
• ${\displaystyle S}$ some ${\displaystyle E}$-valued random variable.

The following are equivalent:^[2]: 40

1. ${\displaystyle S_{n}\to S}$ almost surely.
2. ${\displaystyle S_{n}\to S}$ in probability.
3. ${\displaystyle \mu _{n}\to \mu }$ in the Lévy–Prokhorov metric.
4. ${\displaystyle \{\mu _{n}\}_{n\in \mathbb {N} }}$ is uniformly tight.
5. ${\displaystyle \langle z,S_{n}\rangle \to \langle z,S\rangle }$ in probability for every ${\displaystyle z\in E^{*}}$.
6. There exists a probability measure ${\displaystyle \mu }$ on ${\displaystyle E}$ such that for every ${\displaystyle z\in E^{*}}$

${\displaystyle \mathbb {E} [e^{i\langle z,S_{n}\rangle }]\to \int _{E}e^{i\langle z,x\rangle }\,\mu (\mathrm {d} x).}$

Remarks: Since ${\displaystyle E}$ is separable, item 3 (i.e. convergence in the Lévy–Prokhorov metric) is the same as convergence in distribution ${\displaystyle \mu _{n}\implies \mu }$. If we remove the symmetric-distribution condition:

• in a finite-dimensional setting equivalence is true for all except item 4 (i.e. the uniform tighness of ${\displaystyle \{\mu _{n}\}_{n\in \mathbb {N} }}$),^[2]
• in an infinite-dimensional setting ${\displaystyle 1\iff 2\iff 3}$ is true but ${\displaystyle 6\implies 3}$ does not always hold.^[2]: 37

• Pap, Gyula; Heyer, Herbert (2010). Structural Aspects in the Theory of Probability. Singapore: World Scientific. p. 79.