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?