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1838 V. ANDRIEU, L. PRALY, AND A. ASTOLFIHence, selecting L ≥ 1, there exists a real number ɛ> 0 such thatL −ɛThis implies∣ ( {∣δ τ i ∣(L)∣1 ∣∣∣z ( σ)∣ ∣∣L i ≤ maxL i β τ − σδ ,L L{i∑L −ɛ sup σ≤κ≤τOn the other hand, the function(̂X n ,E 1 ) ↦→ c 01−d∞(n−i−1)(j−1)≥ L1−d∞(n−j) −i ≥ L (j−1) 1−d 0 (n−i−1)1−d 0 (n−j) −i .i∑j=1c 0j=1),|(ˆX τ,j(κ) − e τ,j (κ))| 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑j=1|ˆX j − e j | 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑|(ˆX τ,j(κ) − e τ,j (κ))| 1−d∞(n−i−1)1−d∞(n−j)j=1|ˆX j − e j | 1−d∞(n−i−1)1−d∞(n−j)}}is homogeneous in the bi-limit with weights (r 0 ,r 0 ) and (r ∞ ,r ∞ ) and degrees 1 −d 0 (n − i − 1) = r 0,i + d 0 and 1 − d ∞ (n − i − 1) = r ∞,i + d ∞ (see (3.2)). Hence, byCorollary 2.15, there exists a positive real number c 1 such thati∑c 0j=1|ˆX j − e j | 1−d 0 (n−i−1)1−d 0 (n−j)+ c ∞i∑j=1(5.16) ≤ c 1 H|ˆX j − e j | 1−d∞(n−i−1)1−d∞(n−j)()|(̂X n ,E 1 )| d0+r0,i(r , |(̂X 0,r 0) n ,E 1 )| d∞+r∞,i(r ∞,r ∞).Hence, by Corollary 2.22 (applied in the τ time-scale), there exists c G such that forany L ∗ large enough such that c 1 L ∗−ε ≤ c G , the conclusion holds.Proof of Corollary 5.3. The proof is similar to the previous one with the onlydifference being that, when i and j satisfy 3 ≤ i +2 ≤ j ≤ n, the function s ↦→1−(n−i−1) s1−(n−j) sis strictly decreasing, mapping (−1,1n−1 ) in ( icondition −1 < d ∞ ≤ d 0 < 1n−11 − d ∞ (n − i − 1)1 − d ∞ (n − j)gives the inequalities≥ 1 − d 0(n − i − 1)1 − d 0 (n − j)j−1 ,>n−in+1−jij − 1 .). Moreover theHence (5.16) holds, and by selecting L