# Necessary and sufficient condition for the law of the iterated logarithm in Hilbert space

This is a follow-up of the previous post Law of the iterated logarithm in Hilbert space

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 $$X_1 + \cdots + X_n$$.

As indicated in the previous post by Iosif Pinelis, it is known (see the links in Law of the iterated logarithm in Hilbert space) that if $$X, X_1, X_2, \ldots$$ are iid random vectors in a separable Hilbert space $$(H, \langle \cdot, \cdot \rangle, |\cdot |)$$ with $$E(X) = 0$$ and $$E(|X|^2) < \infty$$, then $$\limsup_{n \to \infty} \frac {|X_1 + \cdots + X_n|}{\sqrt {2n \ln \ln n}} = \sigma$$ almost surely where $$\sigma = \sup \Big \{ \sqrt {E \big (\langle X, f \rangle ^2 \big) } : f \in H, |f| = 1 \Big\}.$$

Now the additional question is about the minimal assumption under which such a property holds. The real law of the iterated logarithm is known to be equivalent to $$\sigma^2 = E(X^2) < \infty$$. Does the law of the iterated logarithm in Hilbert space imply back that $$E(|X|^2) < \infty$$, or only $$\sigma < \infty$$ which seems weaker? Are there necessary and sufficient moment conditions for this law of the iterated logarithm in Hilbert space?

## 1 Answer

As a immediate corollary of the real-valued case, a necessary condition is that for all $$f\in H$$, $$\langle X,f\rangle$$ should be centered and have a finite moment of order two. For $$n\geqslant 3$$, denote $$a_n:=\sqrt{2n\log\log n}$$. If $$\limsup_{n\to+\infty}\lVert S_n\rVert/a_n<+\infty$$ almost surely, so is $$\limsup_{n\to+\infty}\lVert X_n\rVert/a_n<+\infty$$, since $$a_n/a_{n-1}\to 1$$. By the $$0-1$$ law, the latter $$\limsup$$ is almost surely constant (say equal to $$C$$) hence $$\mathbb P\left(\limsup_{n\to+\infty}\left\{\lVert X_n\rVert/a_n>C+1\right\}\right)=0$$. The second Borel-Cantelli lemma thus implies that $$\sum_{n\geqslant 3}\mathbb P\left\{\lVert X_n\rVert/a_n>C+1\right\}$$ converges. Using the fact that the random variables are identically distributed, we get that $$\sum_{n\geqslant 3}\mathbb P\left\{\lVert X_1\rVert/a_n>C+1\right\}<\infty$$. This implies that $$\mathbb E\left[\lVert X_1\rVert^2/LL\left(\lVert X_1\rVert\right)\right]<+\infty$$, where $$L(x)=\max\{1,\log x\}$$ and $$LL(x)=L\circ L(x)$$.

We thus have shown the following:

If $$(X_i)_{i\geq 1}$$ is i.i.d. and such that $$\limsup_{n\to+\infty}\lVert S_n\rVert/\sqrt{2n\log\log n}<+\infty$$, then

1. For each $$f\in H$$, $$\mathbb E\left[\langle X,f\rangle\right]=0$$ and $$\mathbb E\left[\langle X,f\rangle^2\right]$$ is finite.
2. $$\mathbb E\left[\lVert X_1\rVert^2/LL\left(\lVert X_1\rVert\right)\right]<+\infty$$.

Note that in the real-valued case, condition 2. is automatically satisfied as long as 1. is.

These conditions are actually also necessary for almost sure boundedness of $$\limsup_{n\to+\infty}\lVert S_n\rVert/\sqrt{2n\log\log n}$$. This is shown in Theorem 2 of the paper

de Acosta, A.; Kuelbs, J. Some Results on the Cluster Set ,$$C\left(\left\{\frac{S_n}{a_n}\right\}\right)$$ undefined and the LIL. Ann. Probab. 11 (1983), no. 1, 102--122. doi:10.1214/aop/1176993662. https://projecteuclid.org/euclid.aop/1176993662

• Thank you very much, that is most helpful! This is an unexpected and striking equivalence. It is not so clear to me why $\sigma$ is involved rather than only $E(|X|^2)$? – user166870 Oct 12 '20 at 14:02
• At least from the one dimensional case one can see that the lower bound for the $\limsup_n$ of the norm of normalized partial sums is bigger than $\sigma$. I need to think more for an intuitive argument for the upper bound. – Davide Giraudo Oct 15 '20 at 10:00