# pr.probability – 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?