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Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate
Author(s) -
Bardet JeanMarc,
Doukhan Paul,
León José Rafael
Publication year - 2008
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2008.00588.x
Subject(s) - mathematics , series (stratigraphy) , limit (mathematics) , autoregressive model , heteroscedasticity , bilinear interpolation , central limit theorem , asymptotic distribution , estimator , weak convergence , consistency (knowledge bases) , mathematical analysis , econometrics , statistics , discrete mathematics , paleontology , computer security , computer science , asset (computer security) , biology
. We prove uniform convergence results for the integrated periodogram of a weakly dependent time series, namely a strong law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak‐dependence assumptions, the strong consistency and asymptotic normality of Whittle's estimate are established for a large class of models. For instance, the causal θ ‐weak dependence property allows a new and unified proof of those results for autoregressive conditionally heteroscedastic (ARCH)(∞) and bilinear processes. Non‐causal η ‐weak dependence yields the same limit theorems for two‐sided linear (with dependent inputs) or Volterra processes.