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Itô–Nisio theorem

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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 [d] in 1968.[2]

Statement

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Let be a real separable Banach space with the norm induced topology, we use the Borel σ-algebra and denote the dual space as . Let be the dual pairing and is the imaginary unit. Let

  • be independent and symmetric -valued random variables defined on the same probability space
  • be the probability measure of
  • some -valued random variable.

The following is equivalent[2]: 40 

  1. converges almost surely.
  2. converges in probability.
  3. converges to in the Lévy–Prokhorov metric.
  4. is uniformly tight.
  5. in probability for every .
  6. 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

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

References

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