Limit theorems on the Gaussian Wiener chaos
In a recent series of papers (see [69, 82, 86, 87, 90, 95, 98, 103, 110, 111] for the Gaussian case, and [106, 107, 108, 109, 114] for the Poisson case) a set of new results has been established, allowing to obtain neat Central Limit Theorems (CLTs) for sequences of random variables belonging to a fixed Wiener chaos of some Gaussian or Poisson field. The techniques adopted in the above references are quite varied, as they involve stochastic calculus (see [98, 103, 111]), Malliavin calculus (see [82, 95, 110]), Stein’s method (see [86, 87, 90, 106]) and decoupling (see [108, 107, 109]). However, all these contributions may be described as “drastic simplifications” of the method of moments and cumulants (see, for example, [12, 66], as well as the discussion below) which is a common tool for proving weak convergence results for non-linear function- als of random fields.
KeywordsLimit Theorem Polish Space Gaussian Measure Stochastic Calculus Gaussian Case
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