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The application of the Bernstein inequality to estimating the parameter of the first order autoregression. (English. Ukrainian original) Zbl 0923.60036

Theory Probab. Math. Stat. 55, 13-18 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 13-19 (1996).
Let \(H\) be a separable Hilbert space and let \(B\) be a \(\sigma\)-algebra of the Borel subsets of \(X\). The author considers the \(\text{AR}(1)\) process \(x_n=\lambda x_{n-1}+\varepsilon_n\), where \(\lambda\) is an unknown parameter, \(x_k, \varepsilon_k \in (H,B)\) are random elements, \(\varepsilon_k,\;k\geq 0,\) are i.i.d. random variables with \(E\varepsilon_k=0,\;E| \varepsilon_k| ^2=1.\) The least squares estimate of the parameter \(\lambda\) is defined as \[ \lambda_n={{\sum_{k=1}^n(x_k,x_{k-1})}\Bigl/{\sum_{k=1}^n| x_{k-1}| ^2}}. \] Exponential estimates for probability \(P\{\sqrt n| \lambda_n-\lambda| >R\}\) are obtained.

MSC:

60F10 Large deviations
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