Law of the iterated logarithm in Hilbert space

Question

The standard law of the iterated logarithm expresses that if $X_1, X_2, \ldots$ are iid real random variables with mean zero and variance $\sigma^2$, $$ \limsup_{n \to \infty} \frac {X_1 + \cdots + X_n}{\sqrt {2n \ln \ln n}} = \sigma $$ almost surely. Together with the same result for $-X_1, -X_2, \ldots$, the same limit holds true with absolute values around the sum $X_1 + \cdots + X_n$.

Is there a corresponding result when the $X_i$'s take values in a Hilbert space, and the absolute values are replaced by the Hilbert norm?

Answer 1

There are such versions of the law of the iterated logarithm even for independent random vectors in an arbitrary separable Banach space. See e.g. Theorems 4.1 and 4.2.

In the case when the Banach space is a Hilbert one, Theorem 4.1 implies the following:

Theorem 1: Let $X,X_1,X_2,\dots$ be iid random vectors in a separable Hilbert space $(H,\langle\cdot,\cdot\rangle,|\cdot|)$ with $EX=0$ and $E|X|^2<\infty$. Let $S_n:=X_1+\cdots+X_n$. Then $$\limsup_n\frac{|S_n|}{\sqrt{2n\ln\ln n}}=\sigma$$ almost surely, where $$\sigma:=\sup\big\{\sqrt{E\langle X,f\rangle^2}\colon f\in H,|f|=1|\big\}.$$

To deduce Theorem 1 from Theorem 4.1, one only needs to note the following two points:

(i) In view of formula (2.5), $\sup_{x\in K}|x|=\sigma$.

(ii) $E|S_n|\le\sqrt{E|S_n|^2}=\sqrt{nE|X|^2}=o(\sqrt{2n\ln\ln n})$, so that condition (ii) of Theorem 4.1 holds.

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In the case when $H=\mathbb R$, Theorem 1 becomes the law of the iterated logarithm cited in your post.