Let $P$ be a probability measure on a measurable space $(E, \mathcal {E})$, and let $\mathcal {F}$ be a countable collection of measurable functions $f : E \to \mathbb {R}$ which is a Donsker class for $P$ in the sense that if $X_1, \ldots, X_n, \ldots$ are independent random variables with common distribution $P$, the sequence $(\frac {1}{\sqrt n} \sum_{i=1}^n [f(X_i) - E(f(X_i))])_{n \geq 1}$ is uniformly tight over $\mathcal {F}$ and the finite dimensional distributions converge to a (centered) Gaussian process indexed by $\mathcal {F}$. Is it then also true that there is a uniform law of the iterated logarithm, that is $$ \sup_{n} \sup_{f \in \mathcal {F}} \frac {1}{\sqrt {2 n \ln \ln n}} \bigg | \sum_{i=1}^n [f(X_i) - E(f(X_i))] \bigg | < \infty $$ almost surely?
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For such a Donsker class $\mathcal {F}$, the necessary and sufficient condition for the uniform law of the iterated logarithm to hold true is that $E(Z^2/\ln \ln (Z)) < \infty$ where $Z = \sup_{f \in \mathcal {F}} [f(X_1) - E(f(X_1))]$. This is standard material which may be found in the Ledoux-Talagrand 1991 book.
