Proof of the theorem 13 ǫ ≤ Π ǫ, τ + PA ǫ, τ = P max ǫ, τ. ǫ, τ ≤ Π ǫ, τ, where ǫ, τ = P max ǫ, τ ≤ ǫ ≤ c lim sup

Reporting estimate 56 to 54 we get I J , n, q ≤ C X k≤k n ∆b 1 k 1 −qα E |X n ,k | q ≤ X k≤k n σ −2qα n ,k E |X n ,k | q and substituting this to inequality 53 we get Π 1 J , n; ǫ ≤ C 1 ǫ −q 2 −J qα+1−q2 + C 2 X k≤k n σ −2qα n ,k E |X n ,k | q . Thus 51 follows from 10, and condition ii holds.

5.2 Proof of the theorem 13

It suffices to check that 50 and 51 hold. Proof of 50 Define: S n ,τ k = X 1≤ j ≤k X n , j ,τ , S ′ n ,τ k = X 1≤ j ≤k X n , j ,τ − E X n , j ,τ . 57 and A n = ½ max 1≤k≤k n |X k | ≤ τσ 2α n ,k ¾ . Then the we can estimate the probability in 50 by P max 1≤k≤k n |∆ 1 k 1 S n k| ∆b 1 k 1 α ǫ =: Π 1

n, ǫ ≤ Π

1 n, ǫ, τ + PA c n where Π 1

n, ǫ, τ = P max

1≤k≤k n |∆ 1 k 1 S n ,τ k| ∆b 1 k 1 α ǫ . 58 Due to 14 the probability PA c n tends to zero so we need only to study the asymptotics of Π 1

n, ǫ, τ.

Recall the definition 57. Using the splitting ∆ 1 k 1 S n ,τ k = ∆ 1 k 1 S ′ n ,τ k + E ∆ 1 k 1 S n ,τ k, let us begin with some estimate of the expectation term, since X n , j ,τ are not centered. We have E |X n , j ,τ | ≤ E 12 X 2 n , j P 12 |X n , j | τσ 2α n , j 2278 By applying Cauchy inequality we get max 1≤k≤k n |E ∆ 1 k 1 S n ,τ k| ∆b 1 k 1 α ≤ max 1≤k 1 ≤k 1 n P k n ,2:d k 2:d =1 E |X n , j ,τ | ∆b 1 k 1 α ≤ max 1≤k 1 ≤k 1 n ∆b 1 k 1 12 P k n ,2:d k 2:d =1 P|X n ,k | τσ 2α n ,k 12 ∆b 1 k 1 α ≤ max 1≤k 1 ≤k 1 n ∆b 1 k 1 12−α X 1≤k≤k n P|X n ,k | τσ 2α n ,k 12 . Due to 9 and 14 the last expression is bounded by ǫ2 for n ≥ n , where n depends on ǫ and τ. Thus for n ≥ n we have Π 1

n, ǫ, τ ≤ Π

′ 1

n, ǫ, τ, where

Π ′ 1

n, ǫ, τ = P max

1≤k≤k n |∆ 1 k 1 S ′ n ,τ k| ∆b 1 k 1 α ǫ2 59 Since Var X n ,k,τ ≤ E X 2 n ,k,τ ≤ E X 2 n ,k = σ 2 n ,k , using Markov, Doob and Rosenthal inequalities for q 112 − α we get Π ′ 1

n, ǫ, τ ≤

k 1 n X k=1 ǫ2 −q ∆b 1 k −qα E |∆ 1 k S ′ n ,τ k n | q ≤ c k 1 n X k=1 ǫ2 −q ∆b 1 k −qα ∆b 1 k q2 + k n ,2:d X k 2:d =1 E |X n ,k,τ | q ≤ cǫ2 −q k 1 n X k=1 ∆b 1 k q12−α + X 1≤k≤k n σ −2qα n ,k E |X n ,k,τ | q . Note that this estimate holds for each τ 0. Combining all the estimates we get ∀τ 0, lim sup mn→∞ Π 1

n, ǫ ≤ c lim sup

mn→∞ X 1≤k≤k n σ −2qα n ,k E |X n ,k,τ | q . with the constant c depending only on q. By letting τ → 0 due to 16, 50 follows. Proof of 51 Introduce similar definitions ψ n ,τ r, r − and ψ ′ n ,τ r, r − by exchanging variables X n ,k with variables X n ,k,τ and X ′ n ,k,τ := X n ,k,τ − E X n ,k,τ respectively. Similar to the proof of 50 we get that we need only to deal with asymptotics of Π 2 J , n, ǫ, τ, where Π 2 J , n, ǫ, τ = P sup j≥J 2 α j max r∈D j ψ n ,τ r, r − ǫ . 2279 Again we need to estimate the expectation term. We have sup j≥J 2 α j max r∈D j max 1 2:d ≤k