On an estimator of an unknown parameter for a first-order autoregressive procedure \((|\theta|>1)\) (Q2771518)

Let \(H\) be an abstract Hilbert space. Consider the first-order autoregressive model \(x_{n+1}=\theta x_{n}+\varepsilon_{n+1}\), where \(\theta\) is an unknown parameter and innovations \(\{\varepsilon_{n}\}\) are independent, identically distributed \(H\)-valued random elements which have zero mean and satisfy the Cramér condition. Under the assumption NEWLINE\[NEWLINE1+\lambda_0\leq |\theta|\leq L<\infty,NEWLINE\]NEWLINE \(\lambda_0\) and \(L\) being known positive numbers, the authors prove an exponential type upper estimate for the probability \(P\{|\theta|^n|\theta_n-\theta|>R\}\), where \(\theta_n\) is the least squares estimate for \(\theta\). Results of such type lead to a method of constructing confidence intervals for the unknown value of the parameter \(\theta\).