A law of iterated logarithm is established for the maximum likelihood estimator of the unknown parameter of the explosive Gaussian autoregressive process. Outside the Gaussian case, we show that the law of iterated logarithm does not hold, except for a suitable averaging on the maximum likelihood estimator.
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Let (Xn, n ≥ 1) be a sequence of independent identically distributed random variables with a common distribution function F and let Sn = ∑nj = 1, n ≥ 1. When F belongs to the domain of partial attraction of a semistable law with index α, 0 < α < 2, Chover’s form of the law of the iterated logarithm has been obtained for subsequences of (Sn), along with some boundary crossing problems.
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