6. Complementary Results

In this chapter, we prove seven lemmas required in the detailed proofs of the results. The first two are related to the functional estimation seen in Sect. 5.​3.​3.​ Indeed, in that section we explain that in the case where μ ≡ 0, the solution of the SDE is X(t) = K(b _H (t)) where K is a solution of an ODE and thus we assert that proving results enunciated in Remarks 3.​28 and 3.​30 is equivalent to prove them for the fBm. These lemmas give in an explicit manner how the increments of X can be approximated by those of the fBm. The proofs required the use of the modulus of continuity for the fBm and other results proved in Sect. 5.​2.​1.​ The third lemma is a straightforward calculation of the asymptotic variance of the random variable defined as a linear combination of variables of the type \(S_{g,\ell_{i}n}(1)\), used in Sect. 5.​2.​2.​ The fourth lemma is concerned by Sect. 5.​2.​3 where we link \(\hat{H}_{k(n)}\) with \(\hat{H}_{\log }\). In this lemma we proved that the corresponding functionals are equivalent in L ². For this aim we show that the Hermite coefficients for function \(\frac{g_{k(n)}} {k(n)}\) converge to those of function g _log. In the fifth lemma, we prove the almost sure equivalence between the second order increments of X and of σ times the increments of the fBm, referred to in Sect. 5.​3.​1.​ Giving the explicit solution for each of the four models and using the modulus of continuity for the fBm lead to the proof. A similar lemma is then demonstrated in the case where we do hypotheses testing seen in Sect. 5.​3.​2 replacing σ by σ _n and the techniques are the same that for previous lemma. Finally in last and seventh lemma, we get back to functional estimation seen in Sect. 5.​3.​3 where μ is supposed to be null and where we want to prove the stable convergence for a functional of the fBm. This lemma is a step in this progression. More precisely, we prove the L ² equivalence between the looked for functional and its approximation. That is done using regression techniques and straightforward calculus of expectations.