Proof of the proposition 6 Proof of theorem 8 Proof of theorem 3

with T l t := X i∈I cardKi=l Y k∈Ki s k X n ,i , 32 here by cardK we denote the cardinality of the set K. The expression 32 can be further recast as T l t = X K⊂{1,...,d} cardK=l X i∈I Ki=K Y k∈K s k X n ,i = X K⊂{1,...,d} cardK=l Y k∈K s k X i∈I Ki=K X n ,i . It should be clear that X i∈I Ki=K X n ,i = Y k∈K ∆ k u k t k +1 S n Ut , where the symbol Π is intended as the composition product of differences operators. Recalling that s k = t k − b k u k t k ∆b k u k t k + 1, this leads to T l t = X K⊂{1,...,d} cardK=l Y k∈K t k − b k u k ∆b k u k t k + 1 Y k∈K ∆ k u k t k +1 S n Ut . 33 To complete the proof report this expression to the equation 31.

3.3 Rosenthal and Doob inequalities

When applied to our triangular array, Rosenthal inequality for independent non-identically dis- tributed random variables reads E ¯ ¯ ¯ ¯ X 1≤ j ≤n X n , j ¯ ¯ ¯ ¯ q ≤ c X 1≤ j ≤n σ 2 n , j q2 + X 1≤ j ≤n E |X n , j | q , 34 for every q ≥ 2, with a constant c depending on q only. As in [11] we can also extend Doob inequality for independent non-identicaly distributed variables E max 1≤k≤k n |S n k| q ≤ p p − 1 dq E |S n k n | q , 35 for q 1. 4 Finite-dimensional distributions

4.1 Proof of the proposition 6

We have gt, s = lim mn→∞ µ n t ∧ s . 2270 Take p ∈ N, v 1 , . . . , v p ∈ R and t 1 , . . . , t p ∈ [0, 1] d . Note that for any t , s , r ∈ [0, 1] d we have 1{r ∈ [0, t ∧ s ]} = 1{r ∈ [0, t ] ∩ [0, s ]} = 1{r ∈ [0, t ]}1{r ∈ [0, s ]}. 36 Then p X i=1 p X j=1 v i µ n t i ∧ t j v j = p X i=1 p X j=1 v i v j X k≤k n 1{B n k ∈ [0, t i ∧ t j ]}σ 2 n ,k = X k≤k n σ 2 n ,k p X i=1 v i 1{B n k ∈ [0, t i ]} 2 ≥ 0. Since this holds for each n, taking the limit as mn → ∞ gives the positive definiteness of gt , s .

4.2 Proof of theorem 8

Consider the jump process defined as ζ n t = X 1≤k≤k n 1{Bk ∈ [0, t ]}X n ,k . Now for each t |Ξ n t − ζ n t | = X 1≤k≤k n α n ,k X n ,k , where α n ,k = |R n ,k | −1 |R n ,k | − 1{Bk ∈ [0, t ]}. Now |α n ,k | 1, and vanishes if R n ,k ⊂ [0, t ], or R n ,k ∩ [0, t ] = ∅. Actually α n ,k 6= 0 if and only if k ∈ I , where I is defined by 24. Thus E |Ξ n t − ζ n t | 2 = X k∈I α n ,k σ 2 n ,k ≤ X k∈I σ 2 n ,k ≤ d X l=1 ∆b l u l t l + 1. Using 9 we get |Ξ n t − ζ n t | P − → 0, as mn → ∞. We will concentrate now on finite-dimensional distributions of ζ n . Fix t 1 , . . . , t r ∈ [0, 1] d and v 1 , . . . , v r real, set V n = r X p=1 v j ζ n t p = X 1≤k≤k n α n ,k X n ,k , where α n ,k = r X p=1 v p 1{Bk ∈ [0, t p ]}. 2271 Now using 36 we get b n := E V 2 n = X k≤k n α 2 n ,k σ 2 n ,k = X k≤k n X p X q v p v q 1{Bk ∈ [0, t p ]}1{Bk ∈ [0, t q ]}σ 2 n ,k = X p X q v p v q µ n t p ∧ t q . Letting mn to to infinity and using assumption 5, we obtain b n −−−−−→ mn→∞ X p X q v p v q µt p ∧ t q = E X v p Gt p 2 =: b. If b = 0, then V n converges to zero in distribution since E V 2 n tends to zero. In this special case we also have P p v p Gt p = 0 almost surely, thus the convergence of finite dimensional distributions holds. Assume now, that b 0. For convenience put Y n ,k = α n ,k X n ,k and v = P p P q v p v q . Since Y 2 n ,k ≤ vX 2 n ,k , Y n ,k satisfies the condition of infinitesimal negligibility. For mn large enough to have b n b2, we get 1 E V 2 n X 1≤k≤k n E Y 2 n ,k 1{|Y n ,k | 2 ǫ 2 E V 2 n } ≤ 2v b X 1≤k≤k n E X 2 n ,k 1 |X n ,k | 2 bǫ 2 2v . Thus Lindeberg condition for V n is satisfied and that gives us the convergence of finite dimensional distributions. 5 Tightness results

5.1 Proof of theorem 3

We will use theorem 14. Using Doob inequality we have P sup t ∈[ 0,1] d |Ξ n t | a = Pmax k≤k n |S n k| a ≤ a −2 E S n k n 2 = a −2 → 0, as a → ∞, thus condition i is satisfied. For proving ii note that due to the definitions of v , v + and v − we can write Ξ n v − Ξ n v + = l X i=1 Ξ n v + w i−1 − Ξ n v + w i Ξ n v − Ξ n v − = l X i=1 Ξ n v − w i−1 − Ξ n v − w i 2272 where l is the number of odd coordinates in 2 j v , w = 0, w i has 2 − j in the first i odd coordinates of 2 j v , and zero in other coordinates. So the condition ii holds provided one proves for every ǫ 0 lim J →∞ lim sup n→∞ Π − J , n; ǫ = 0, 37 lim J →∞ lim sup n→∞ Π + J , n; ǫ = 0, 38 where Π − J , n; ǫ := P sup j≥J 2 α j max r∈D j 0≤ℓ≤2 j |Ξ n

r, s

ℓ − Ξ n r − , s ℓ | ǫ , 39 Π + J , n; ǫ := P sup j≥J 2 α j max r∈D j 0≤ℓ≤2 j |Ξ n r + , s ℓ − Ξ n

r, s

ℓ | ǫ , 40 with D j = {2l − 12 − j ; 1 ≤ l ≤ 2 j−1 }, r − = r − 2 − j , ℓ = l 2 , . . . , l d , 2 j = 2 j , . . . , 2 j vector of dimension d − 1 and s ℓ = ℓ2 − j . We prove only the 37, since the proof of 38 is the same. Denote by v = r, s ℓ , and v − = r − , s ℓ . From 32 we have Ξ n

r, s

ℓ − Ξ n r − , s ℓ = S n Uv − S n Uv − + d X l=1 T l v − T l v − . To estimate this increment we discuss according to following configurations Case 1. u 1 r = u 1 r − . Consider first the increment T 1 v − T 1 v − and note that by 33 with l = 1, T 1 v = X 1≤k≤d v k − b k u k v k ∆b k u k v k + 1 ∆ k u k v k +1 S n Uv. Recall the notation 5 and note that by definition v 2:d = v − 2:d with U v = Uv − . Thus all terms indexed by k ≥ 2 disappear in difference T 1 v − T 1 v − . This leads to the factorisation T 1 v − T 1 v − = r − r − ∆b 1 u 1 r + 1 ∆ 1 u 1 r+1 S n Uv. 41 For l ≥ 2, T l v is expressed by 33 as T l v = X 1≤i 1 ···i l ≤d v i 1 − b i 1 u i 1 v i 1 ∆b i 1 u i 1 v i 1 + 1 . . . v i l − b i l u i l v i l ∆b i l u i l v i l + 1 ∆ i 1 u i1 v i1 +1 . . . ∆ i l u il v il +1 S n Uv. In the difference T 1 v − T 1 v − all the terms for which i 1 ≥ 2 again disappear and we obtain T l v − T l v − = r − r − ∆b 1 u 1 r + 1 X 1i 2 ···i l ≤d v i 2 − b i 2 u i 2 v i 2 ∆b i 1 u i 1 v i 1 + 1 . . . v i l − b i l u i l v i l ∆b i l u i l v i l + 1 ∆ 1 u 1 r+1 ∆ i 2 u i2 v i2 +1 . . . ∆ i l u il v il +1 S n Uv. 42 2273 Since u 1 r = u 1 r − , we have b 1 u 1 r ≤ r r − b 1 u 1 r + 1, thus r − r