Exponential estimate for the law of the iterated logarithm in Banach space (Q1905316)

Let \(\xi_k\), \(k = 1,2, \dots\), be i.i.d. random variables with values in a separable Banach space \(X\) with the norm \(|x |\), \(x \in X\). Assume that \(E \xi_k = 0\) and \(E |f (\xi_k) |^2 < \infty\) for every continuous functional \(f(x)\), \(x \in X\). The main result of the paper is the following inequality \[ P(u) = P \left( \sup_{n \geq 1} \left[ \bigl( 2n \log \log (n + e) \bigr)^{- 1/2} \left |\sum^n_{k = 1} \xi_k \right |\right] > u \right) \leq D \exp \{- Cu^{\min (q,2)}\} \] if \[ \int^1_0 H^{\max (1/q, 1/2)} (p, \varepsilon) d \varepsilon < \infty, \quad 1/p + 1/q = 1, \] where \(H(p, \varepsilon)\) is the entropy constructed with the help of the random variables and the set of extreme points of the dual space \(X^*\), \(C\) and \(D\) are some positive constants. In particular, in the one-dimensional case it is also proved that for any \(q > 1\) there exists a sequence of i.i.d. random variables with \[ p \bigl( |\xi|> u \bigr) \leq \exp \{-Ku^q\}, \quad u > 1, \] such that \[ P(u) \geq D \exp \bigl \{-C \min (u^q, u^2 \log \log u) \bigr\}, \quad u \geq 4, \] where \(C,D,K\) are some positive constants.