When the cone condition fails (Q2507984)

Let \((X,\mathcal A,\mu,\tau)\) be an ergodic dynamical system and \(T\) the isometry associated with \(\tau\) by the relation \(Tf=f\circ \tau\). Let \(Z=\{Z(h),\;h\in L^2(\mu)\}\) be the canonical Gaussian centered process on a suitable basic probability space \((\Omega,\mathcal K,\mathbb P)\), with corresponding mean value operator \(\mathbb E\). It is well-known that \(Z\) is characterized by the covariance function \(\mathbb E Z(h)Z(h')=(h,h')\), \(h,h'\in L^2(\mu)\). Finally, recall that a countable set \(A\) in \(L^2(\mu)\) is said to be Gaussian bounded if \(\mathbb E\sup_{h\in A}|Z(h)|<\infty\). For suitable sequences \((n_u)_{u\geq 1}\) and \((\ell_u)_{u\geq 1}\) of integers depending on a fixed real number \(\theta>\sqrt 2\), consider the sequence of the associated moving averages \[ B^T_{\theta,u} h=\frac 1{\ell_u}\sum^{n_u+\ell_u-1}_{\ell=n_u} T^\ell h,\quad h\in L^2(\mu). \] The main result of the paper is as follows. Let \(\{f_n,\;n\geq 0\}\subset L^\beta(\mu)\) with \(\beta>\theta^2\). Assume that (i) \(\lim_{u\to \infty}B^T_{\theta,u}h\overset{\text{a.s.}}{=}\int hd\mu\) for \(h=f_mf_n\), \(n,m\geq 1\), (ii) \(\{f_n,\;n\geq 0\}\) is a Gaussian bounded set. Then, for any \(R\geq 0\), \[ \mu\Big \{\sup_{n\geq 0}|f_n|>R\Big\} \geq 4\mathbb P\Big\{\sup_{n\geq 0}|Z(f_n)|\geq \lambda_0R\Big\}\leq 8e^{-R^2/E(f)^2}, \] where \(E(f)=\|\sup_{n\geq 0}|Z(f_n|\,\|_\Phi<\infty\), and \(\|\cdot\|_\Phi\) is the Orlicz norm with respect to the Young function \(\Phi(x)=e^{x^2}-1\), and \(\lambda_0\) is such that \(\mathbb P\{|\mathcal N(0,1)|\geq \lambda_0\}=3/4\). For related work (where unlike the present paper the so-called cone conditions holds), see \textit{A. Bellow, R. Jones} and \textit{J. Rosenblatt} [Ergodic Theory Dyn. Syst. 10, No. 1, 43--62 (1990; Zbl 0674.60035)].