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Étude de l’estimateur du maximum de vraisemblance dans le cas d’un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence. (Asymptotic behaviour of maximum likelihood estimator in an autoregressive process: consistency, asymptotic distribution and expansion, rate of convergence). (French) Zbl 0714.60014

The authors consider the group G of all invertible matrices \(g=\begin{pmatrix} A(g) & b(g)\\ 0&1 \end{pmatrix}\), where \(b(g)\in R^ d\) is a column, a norm \(\|.\|\) in \(R^ d\), a probability \(\mu\) on G for which the corresponding norm \(\| A(g)\|\) is \(\mu -a.e.<1\), a sequence \(g_ n\), \(n\geq 1\), of independent \(\mu\)-distributed random variables, a fixed \(z\in R^ d\), the distribution \(\nu\) of \(\sum_{k\geq 1}A(g_ 1)...A(g_{k-1})b(g_ k)\), \(\eta: G\to [1,\infty)\), constants \(c,\gamma,\tau >0\), \(\alpha\in (0,1]\), a function \(F: G\times R^ d\to R\) satisfying \[ | F(g,x)-F(g,y)| \leq \eta (y)\| x- y\|^{\gamma}(1+\| x\|^{\tau}+\| y\|^{\tau}) \] and \[ | F(g,x)| \leq \eta (g)(\| x\|^{\gamma +\tau}+1), \] \(\begin{pmatrix} Y_ n\\ 0 \end{pmatrix} =g_ n...g_ 1\begin{pmatrix} z\\ 0\end{pmatrix}\) for \(n\geq 0\), \(S_ n=\sum^{n}_{1}F(g_ k,Y_{k-1})\), \(e=\int \int Fd\mu d\nu\) and impose \[ (*)\quad \int \eta (g)^ 5(1- \| A(g)\|^{\alpha})^{-5(\gamma +\tau)/\alpha}\exp (c\| b(g)\|^{\alpha})d\mu (g)<\infty. \] First they prove that \(\sigma^ 2=\lim_{n} n^{-1}E((S_ n-ne)^ 2)\) exists, does not depend on z, and, under a supplementary condition on F, is positive, and that \[ d(n^{-1/2}\sigma^{-1}(S_ n-ne),N)\leq Cn^{-1/2}(1+\| x\|^{\gamma \beta}\exp (\lambda \| x\|^{\alpha}) \] with constants \(C,\lambda >0\), \(\beta >1\), where d is the supremum of the modulus of the difference of the distribution functions and N is standard normal. Replacing (*) by two other conditions, each containing \(m\geq 2\) at the exponent, and requiring \(\| A(g)\| \leq \rho <1\mu\)-a.e., they prove also that \[ \sup_{n} E(| n^{- 1/2}\sum^{n}_{1}(F(g_ k,Y_{k-1})-e)|^ m)\leq C(1+\| x\|^{m(\gamma +\tau)}). \] In the proofs they use the operators \((U_ tf)(x)=\int e^{itF(g,x)}f(g,x)d\mu (g)\) and arrange a Banach space on which \(U_ t\) are quasicompact etc.
Then the authors consider the particular case, corresponding to the title, of a nonrandom \(A(g_ n)=A_{\theta}=\left( \begin{matrix} \theta \\ 1\quad 0\end{matrix} \right)\), where \(\theta =(\theta_ 1,...,\theta_ d)\), \(b(g_ n)=\epsilon_ ne_ 1\), \(e_ 1=(1,0,...,0)'\), E \(\epsilon\) \({}_ n=0\), \(\max | u_ i| <1\), where \(u_ i\) are the roots of \(u^ d=\sum \theta_ iu^{d-i}\). It follows that, if \(E(\log^+| \epsilon_ k|)<\infty\), \(\nu\) is the unique invariant distribution of \(P_{\theta}\), where \((P_{\theta}f)(x)=E(f(A_{\theta}x+\epsilon_ ne_ 1)).\) The previous results are applied, and for \(m\geq 4\), \[ P(n^{- 1/2}\sum^{n}_{1}(F(\epsilon_ k,Y_{k-1})-e)>((m-3)\log n)^{1/2})=o(n^{-(m-3)/2}) \] is established. Furthermore, supposing that \(\epsilon_ n\) have a positive density f, tending to 0 at \(\infty\), with a derivative satisfying \(\int | f'|^ rf^{1-r}<\infty\) with an \(r\geq d+1\), and also a finite r-moment, the authors prove, denoting by \({\hat \theta}{}_ n\) the maximum likelihood estimator of \(\theta\) in a \(P_{\theta}\)-Markov chain starting from z, several “statistical results”. These are: the convergence in distribution of \(n^{1/2}({\hat \theta}_ n-\theta)\) to an \(N(0,\sigma^ 2)\), the convergence of the corresponding “absolute k-moments” (for all k), \[ P(\| {\hat \theta}_ n-\theta \| >\rho)\leq A_ 1\exp (-A_ 2n)\text{ for } every\quad \rho >0, \] and \[ P(n^{1/2}\| {\hat \theta}_ n-\theta \| >B \log^{1/2}n)=O(n^{-1}). \] Supposing the existence of \(f^{(p+1)}\) and the validity of other conditions in which a k appears, and writing \[ n^{1/2}({\hat \theta}_ n-\theta)=h_ 1+...+n^{-(p-1)/2}h_ p+n^{-p/2}\rho (n), \] the authors show that \[ P(\| \rho (n)\| >((k-2)\log n)^{(p+1)/2})=O(n^{-(k-2)/2}) \] and also, for \(p=1\), that \[ d(n^{1/2}({\hat \theta}_ n- \theta),N(0,\sigma^ 2))\leq Cn^{-1/2}\log^{1/2}n. \] The uniformity, in z (and \(\theta\)) on compacts, of “the results” is established. Finally, a more precise result is announced.
Reviewer: I.Cuculescu

MSC:

60F05 Central limit and other weak theorems
62F10 Point estimation
60J05 Discrete-time Markov processes on general state spaces
60G50 Sums of independent random variables; random walks
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