Sample path large deviations for squares of stationary Gaussian processes (Q2845221)

The aim of this paper is to provide a large deviation principle (LDP) for random functions of the type NEWLINE\[NEWLINE Z_n(t)=\frac{1}{n}\sum_{k=1}^{[nt]}X^2_k, NEWLINE\]NEWLINE and the associated polygonal line NEWLINE\[NEWLINE \tilde{Z}_n(t)=Z_n(t)+(t-\frac{[nt]}{n})X^2_{[nt]+1}, NEWLINE\]NEWLINE where \(\{X_k\}_k\) is a stationary Gaussian process having continuous positive spectral density \(f\) defined on the torus \(T=[-\pi,\pi]\).NEWLINENEWLINELet \(bv([0,1],{\mathbb R})\) be the space of bounded variation real functions. Let \({\mathcal M}([0,1])\) be the set of bounded measures on \([0,1]\) endowed with the weak topology. We can identify \(bv([0,1],{\mathbb R})\) with \({\mathcal M}([0,1])\): \(\mu_h\) in \({\mathcal M}([0,1])\) corresponds to \(h\) in \(bv([0,1],{\mathbb R})\) characterized by \(\mu_h([0,t])=h(t)\). Up to this identification, the topological dual of \(bv([0,1],{\mathbb R})\) is the set \({\mathcal C}([0,1])\) (the space of continuous functions on \([0,1]\)). We end \(bv([0,1],{\mathbb R})\) with the \(w^*\)-topology written by \(\sigma\), i.e., the topolgy induced by \({\mathcal C}([0,1])\) on \({\mathcal M}([0,1])\). Now we can carry the LDP to the random functions \(Z_n\) and \(\tilde{Z}_n\).NEWLINENEWLINETheorem 2.2. The families of random functions \(Z_n\) and \(\tilde{Z}_n\) satisfy an LDP on the space \((bv([0,1],{\mathbb R}),\sigma)\) with speed \(n\) and rate function \(\Phi\), which is known in an explicite form.NEWLINENEWLINEThe author also shows the moderate LDP for both random families.