Rate of Convergence in the Functional Law of the Iterated Logarithm for Non-standard Normalizing Factors A.V. Bulinskii, M.A. Lifshits Let W (t, ω) be a Brownian motion defined on a probabilistic space (Ω, F, P ) for t ≥ 0. Consider the space C[0, 1] equipped with uniform norm and the related uniform metric ρ. Define ”Strassen ball” K as the set of all absolutely continuous functions x(·) defined on [0, 1] such that x(0) = 0 and Z 1 x(t) ˙ 2 dt ≤ 1. 0

Define the random functions XT (t, ω) = W (T t, ω)/T 1/2 ,

t ∈ [0, 1], T ≥ 0, ω ∈ Ω. 1/2

Let LT = 2 log log T for T ≥ 3 and ZT = XT /LT . The classical Strassen functional law of the iterated logarithm states that with probability one the family ZT , T ≥ 3, is relatively compact and the set of its limit points in the space C[0, 1] coincides with K. Consider now more general family of random functions YT (t, ω) = XT (t, ω)/ϕT , T ≥ 3,

(1)

where ϕ is a positive nondecreasing function on R+ such that ϕT → ∞ for T → ∞. According to the work of A.V.Bulinskii, the set K will be the limit set for the family {YT } in the space C[0, 1] with non-standard normalization ϕ iff 1/2

lim inf ϕT /LT T →∞

= 1.

(2)

Whenever Strassen theorem states that for each positive ε and a.e. ω ∈ Ω ρ(ZT , K) := inf{ρ(ZT , x) : x ∈ K} ≤ ε T ≥ T (ω, ε), the next more difficult question relates to the fitting of ZT in the ε- neighbourhoods of K with ε vanishing when T tends to infinity. In our work complete description is given for the family of random functions YT with 1/2 non-standard normalizing functions ϕ to the set K. Denote δT = ϕT − LT and define  −1/2 1/2 −1/6   |δT |LT , −LT ≤ δT ≤ −LT , −2/3 −1/6 ψT = LT , |δT | ≤ LT ,   −1/2 −3/2 −1/6 δT ϕT , δT ≥ LT . 1

Our principal result is as follows. Theorem 1. Let ϕ be a nondecreasing right-continuous function and condition (2) holds. Then, with probability one, 0 < C1 ≤ lim sup ρ(YT , K)/ψT ≤ C2 < ∞. T →∞

Moreover, one can put C1 ≥ 0.05 and C2 ≤ 11.

2