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## Dimension of a Gaussian random variable

Remember a Gaussian variable $$X$$ in a Banach space $$B$$ is a r.v. such that for each $$\xi \in B^\star,$$ the real r.v. $$\xi(X)$$ is a symmetric Gaussian variable. There are two notions of moments: strong moments, $$m_p(X) = (\E\|X\|^p)^{1/p},$$ and weak moments, $$\sigma_p(X) = \sup \{ (\E\xi(X)^p)^{1/p} \, \mid \, \|\xi\|_{B^\star} \leq 1 \}.$$ [...]

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### Banach or not Banach?

Academics / Mathematics : E. Kowalski's Blog (4 years ago)

In the basic theory of normed vector spaces and Banach spaces, there are a number of important theorems involving one or two such spaces, and each of them may, or may not, need to be complete (i.e., really a Banach space, and not on... Read Post

### How well do moments determine a distribution?

Academics : The Endeavour (6 months ago)

If two random variables X and Y have the same first few moments, how different can their distributions be? Suppose E[Xi] = E[Yi] for i = 0, 1, 2, … 2p. Then there is a polynomial P(x) of degree 2p such that |F(x) – G(x)| ? 1/P(x) wh... Read Post

### Comparing products of gaussian moments with one gaussian moment

Academics / Mathematics : Chapter Zero (3 years ago)

I found this result while trying to prove that the moments of a sum of scaled chi-squared random variables grow like : Let be a standard Gaussian random variable. Then for any positive integers (I may have missed some way in which t... Read Post