Summary
Let {μ n (t),t ∈ [O,T n ]} be the sequence of trends andX n − μ n (t) is a stationary Gaussian process having mean zero. This paper establishes the exact asymptotic behaviour for the probabilities\(\Pr \mathop {(\max |X_n (t)|)}\limits_{[O, T_n ]} \leqq u_n ),\mathop {\lim }\limits_{x \to \infty } T_n = + \infty \), in the situation when the {μ n (t)} are not asymptotically negligible and essentially contribute in the approximating distribution. The results are used to study the optimal rates of convergence for nonparametric density estimation when the closeness is measured inL ∞ norm.
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Konakov, V.D. Extrema of some Gaussian processes with large trends and density estimation inL ∞ norm. Probab. Th. Rel. Fields 86, 277–304 (1990). https://doi.org/10.1007/BF01208254
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DOI: https://doi.org/10.1007/BF01208254
Keywords
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- Density Estimation
- Mathematical Biology