Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations
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We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors (S_n)_n ∈ = (n^-1(∑_k=1^n X_k, ∑_k=1^n X_k^2))_n ∈. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence (_n)_n ≥ 2 = (n^-1(∑_k=1^n X_k^2, ∑_k=2^n X_k X_k+1))_n ≥ 2. Via Contraction Principle, we provide explicit rate functions for the sequences (n^-1∑_k=1^n X_k)_n ∈, (n^-1∑_k=1^n X_k^2)_n ≥ 2 and (n^-1∑_k=2^n X_k X_k+1)_n ≥ 2, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.
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