## 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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