Asymptotic theory for fractional regression models via Malliavin calculus
The authors study the asymptotic behavior as \(n\to\infty\) of the sequences \[ S_n= \sum^{n-1}_{i=0} K(n^\alpha B^{H_1}_i)(B^{H_2}_{i+1}- B^{H_2}_i), \] \[ \langle S\rangle_n= \sum^{n-1}_{i=0} K^2(n^\alpha B^{H_1}_i), \] where \(B^{H_1}\) and \(B^{H_2}\) are two independent fractional Brownian motions with Hurst parameters \(H_1\) and \(H_2\), \(K\) is a nonnegative kernel satisfying \[ \int_{\mathbb{R}} K^2(y)\,dy= 1\quad\text{and}\quad \int_{\mathbb{R}} yK(y)\,dy= 0 \] with (possibly) some additional properties, \(\alpha> 0\). The main results are as follows. The authors prove the limits in distribution \[ \lim_{n\to\infty} n^{\alpha+ H_1-1}\langle S\rangle_n = \int_{\mathbb{R}} K^2(y)\,dy L^{H_1}(1, 0), \] \[ \lim_{n\to\infty} n^{\alpha+ H_1-1} S_n= \int_{\mathbb{R}} K^2(y)\,dy W_{L^{H_1}(1,0)}\;\text{if }\alpha< H_1-1, \] where \(L^{H_1}(1,0)\) is the local time of the functional motion \(B^{H_1}\) and \(W\) a Brownian motion independent from \(B^{H_1}\). Also, they prove a theorem on stable convergence of the sequence \(S_n\). The authors use different mathematical techniques and, in particular, classical Itô integration and elements of Malliavin calculus. Reviewer's remark: In the main results, the parameter \(\alpha\) is not specified. (In Lemma 3, the condition \(\alpha- 4H_2+ H_1+ 2> 0\) is used.) For this reason I found it difficult to understand the results.
scientific article