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An empirical central limit theorem for intermittent maps. (English) Zbl 1203.60039

Probab. Theory Relat. Fields 148, No. 1-2, 177-195 (2010); erratum ibid. 155, No. 1-2, 487-491 (2013).
Let \((X_i)_{i\in\mathbb{Z}}\) be a stationary sequence of real-valued random variables with common continuous distribution function \(F\) and let \(F_n\) be the empirical distribution function defined by \((X_i)\). Let \({\mathcal M}_l= \sigma(X_i, i\leq l)\) and \({\mathcal M}_{-\infty}= \bigcap_{i\in\mathbb{Z}}{\mathcal M}_i\). Furthermore, let \(P\) be the law of \(X_0\) and \(P_{(X_i,X_j)}\) be the law of \((X_i, X_j)\), \(P_{X_k|{\mathcal M}_l}\) be the conditional distribution of \(X_k\) given \({\mathcal M}_l\) and \(P_{(X_i,X_j)|{\mathcal M}_l}\) be the conditional distribution of \((X_i, X_j)\) given \({\mathcal M}_l\). Put \(f_t= I_{[-\infty,t]}\) and \(f^{(0)}_t= f_t- P(f_t)\) and define the coefficient \[ \begin{split} \beta_2(k)= \max\Biggl\{E\sup_{t\in\mathbb{R}}\,|P_{X_{k+1}|{\mathcal M}_l}(f_t)- P(t_t)|,\\ E\sup_{i> j\geq k+l}\;\sup_{(s,t)\in\mathbb{R}^2} |P_{(X_i,X_j)|{\mathcal M}_l}(f^{(0)}_t\otimes f^{(0)}_s)- P_{(X_i, X_j)}(f^{(0)}_t\otimes f^{(0)}_s)|\Biggr\}.\end{split} \] (Cf. Many examples of \(\beta_2(k)\) are shown in J. Dedecker and C. Prieur’s paper [Stochastic Processes Appl. 117, No. 1, 121–142 (2007); erratum ibid. 119, No. 6, 2118–2119 (2009; Zbl 1117.60035)].)
In this paper, the author proves the following: if \(\beta_2(n)= O(n^{-1-\delta})\) for some \(\delta> 0\), then \(\sqrt{n}(F_n- F)\) converges in distribution in \(\ell^\infty(\mathbb{R})\) to a centered Gaussian process \(G\) whose covariance function is given by \[ \text{Cov}(G(s), G(t))= \sum_{k\in\mathbb{Z}} \text{Cov}(I_{X_0\leq t}, I_{X_k\leq s}), \] and whose sample paths are a.s. uniformly continuous with respect to the pseudo-metric \(d(x, y)=|F(x)- F(Y)|\). The result is applied, among others, to weak convergence of the iterates of the intermittent map \(T_\gamma\), which is defined by \(T_\gamma(x)= x(1+ 2^\gamma x^\gamma)\) if \(x\in[0, 1/2)\); \(= 2x- 1\) if \(x\in[1/2,1]\), with \(\gamma< 1/2\). In the case \(\gamma= 1/2\), convergence in distribution of the finite-dimensional marginals is also considered.

MSC:

60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
37E05 Dynamical systems involving maps of the interval

Citations:

Zbl 1117.60035
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References:

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