Strong approximation of the empirical distribution function for absolutely regular sequences in \({\mathbb R}^d\). (English) Zbl 1293.60043

Dedecker, Jérôme; Rio, Emmanuel; Merlevède, Florence

Electron. J. Probab. 19, Paper No. 9, 56 p. (2014).

The problem to find optimal rates of lower and upper bounds for the strong approximation of the empirical processes associated to some stationary sequences has a long history. This paper is the one which challenges the problem based on absolutely regular sequences in \(\mathbb{R}^d\). The absolute regularity is one of the weakly dependent conditions.
More precisely, let \((X_i)_{i\in\mathbb{Z}}\) be a stationary absolutely regular sequence in \(\mathbb{R}^d\) with common distribution function \(F\) and define the empirical process of \((X_i)_{i\in\mathbb{Z}}\) by \[ R_X(s,t)= \sum_{1\leq k\leq t} (I_{X_k\leq s}- F(s)),\quad s\in\mathbb{R}^d,\;t\in\mathbb{R}^+. \] Put \[ \Gamma_X(s,s',t,t')= \min(t,t') \sum_{k\in\mathbb{Z}} \text{Cov}(I_{X_0\leq s}, I_{X_k\leq s'}),\quad (s,s')\in \mathbb{R}^{2d},\;t\in\mathbb{R}^+\times \mathbb{R}^+, \] which may be defined if \(\beta_n= O(n^{1-p})\) for some \(p\in]2,3]\). As for the upper bound, among others, the authors prove the following: One can construct a centered Gaussian process \(K_X\) with the covariance function \(\Gamma_X\) in such a way that \[ \sup_{s\in\mathbb{R}^d,k\leq n}|R_X(s,k)- K_X(s,k)|= O(n^{{1\over p}}(\log n)^{\lambda(d)+\varepsilon+{1\over p}})\quad\text{a.s., for any }\varepsilon> 0, \] where \[ \lambda(d)= \Biggl({3d\over 2}+ 2- {2+d\over 2p}\Biggr) I_{p\in ]2,3[}+ \Biggl(2+ {4d\over 3}\Biggr) I_{p=3}. \] As for the lower bound, they give an example of a stationary absolutely regular Markov chain with state space \([0, 1]\) and \(\beta_k=O(k^{1-p})\) \((p>2)\) having the property that for any construction of a sequence \((G_n)_{n>0}\) of continuous Gaussian processes on \([0, 1]\) \[ \limsup_{n\to\infty} (n\log n)^{-{1\over p}} \sup_{s\in (0,1]} |R_X(s,n)- G_n(s)|>0\quad\text{a.s.} \] holds.

Reviewer: Ken-ichi Yoshihara (Yokohama)

MSC:

60F17	Functional limit theorems; invariance principles
60G10	Stationary stochastic processes

Keywords:

strong approximation; Kiefer process; empirical process; stationary sequence; absolutely regular; Markov chain