On the moment convergence rates of LIL in Hilbert space

Let { X , X n ; n ≥ 1 } be a sequence of i.i.d. random variables taking values in a real separable Hilbert space ( H , ‖ ⋅ ‖ ) with mean zero and covariance Σ and set S n = ∑ k = 1 n X k , n ≥ 1 . Let σ i 2 , i ≥ 1 be the eigenvalues of Σ arranged in non-increasing order and take into account the multiplicities. Let l be the dimension of the corresponding eigenspace of largest eigenvalue σ 2 = σ 1 2 . Let log x = ln ( x ∨ e ) , x ≥ 0 . This paper studies the precise rates for ∑ n = 1 ∞ ( log n ) b n 3 / 2 E { ‖ S n ‖ − σ ε 2 n log log n } + . We show that when l > 1 and b > − 1 , E ( ‖ X ‖ 2 ( log ‖ X ‖ ) 3 b + 3 / log log ‖ X ‖ ) < ∞ implies lim ε ↘ 1 + b ( 1 − 1 + b ε 2 ) ł − 1 2 ∑ n = 1 ∞ ( log n ) b n 3 / 2 E { ‖ S n ‖ − σ ε 2 n log log n } + = 1 2 ( b + 1 ) Γ − 1 ( l / 2 ) K ′ ( Σ ) Γ ( ( l − 1 ) / 2 ) , where Γ ( ⋅ ) is the Gamma function and K ′ ( Σ ) = ∏ i = l + 1 ∞ ( ( σ 2 − σ i 2 ) / σ 4 ) − 1 / 2 .