Strong consistency and CLT for the random decrement estimator (Q606316)

Let \(\{X_{i},i\geq 0\}\) be a scalar stationary ergodic Gaussian sequence with zero mean and variance \(\sigma ^2\) where \(\sigma >0\). Let \(d\) be a fixed integer and \(\Delta \) be a convenient domain in \(\mathbb{R}^d\). Denote by \(( \tau _k,k\in \mathbb{Z}) \) the stationary time process defined for \(k>0\) as follows: \[ \tau _{1}=\inf \{j\geq 0:( X_j,X_{j+1},\dots,X_{j+d-1}) \in \Delta \}, \] \[ \tau _{k+1}=\inf \{j>\tau _{k}:( X_j,X_{j+1},\dots,X_{j+d-1}) \in \Delta \}. \] For \(k\leq 0\), it is defined as the stationary extension of this sequence. Let \(l(n)\) be its counting function, that is, the number of \(\tau _k\)'s in the set \(\{0,1,\dots,n-1\}\). Set \[ D_n( j) =\frac{1}{n}\sum_{k=1}^n X_{\tau _{k}+j}, \] \[ \overline{D}_n( j) =\frac{1}{l( n) } \sum_{k=1}^{l(n)} X_{\tau _{k}+j}. \] The authors establish a Law of Large Numbers and a Central Limit Theorem for \(D_n( j) ,n>0\) and \(\overline{D}_n(j),n>0\), for any fixed \(j\). The limit is expressed in terms of some conditional expectation. In some important cases for applications, this conditional expectation is calculated.