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The convergence properties of the empirical characteristic process $Y_n(t) = n{\textasciicircum}\{1/2\}(c_n(t) - c(t))$ are investigated. The finite-dimensional distributions of $Y_n$ converge to those of a complex Gaussian process $Y$. First the continuity properties of $Y$ are discussed. A class of counterexamples is presented, showing that if the underlying distribution has low logarithmic moments then $Y$ is almost surely discontinuous, and hence $Y_n$ cannot converge weakly. When the underlying distribution has high enough moments then $Y_n$ is strongly approximated by suitable sequences of Gaussian processes with specified rate-functions. The approximation is based on that of Komlos, Major and Tusnady for the empirical process. Convergence speeds for the distribution of functionals of $Y_n$ are derived. A Strassen-type log log law is established for $Y_n$, and supremum-functionals on the appropriate set of limit points are explicitly computed. The technique throughout uses results from the theory of the sample function behaviour of Gaussian processes.

@article{csorgo_limit_1981, title = {Limit {Behaviour} of the {Empirical} {Characteristic} {Function}}, volume = {9}, issn = {0091-1798, 2168-894X}, url = {https://projecteuclid.org/journals/annals-of-probability/volume-9/issue-1/Limit-Behaviour-of-the-Empirical-Characteristic-Function/10.1214/aop/1176994513.full}, doi = {10.1214/aop/1176994513}, abstract = {The convergence properties of the empirical characteristic process \$Y\_n(t) = n{\textasciicircum}\{1/2\}(c\_n(t) - c(t))\$ are investigated. The finite-dimensional distributions of \$Y\_n\$ converge to those of a complex Gaussian process \$Y\$. First the continuity properties of \$Y\$ are discussed. A class of counterexamples is presented, showing that if the underlying distribution has low logarithmic moments then \$Y\$ is almost surely discontinuous, and hence \$Y\_n\$ cannot converge weakly. When the underlying distribution has high enough moments then \$Y\_n\$ is strongly approximated by suitable sequences of Gaussian processes with specified rate-functions. The approximation is based on that of Komlos, Major and Tusnady for the empirical process. Convergence speeds for the distribution of functionals of \$Y\_n\$ are derived. A Strassen-type log log law is established for \$Y\_n\$, and supremum-functionals on the appropriate set of limit points are explicitly computed. The technique throughout uses results from the theory of the sample function behaviour of Gaussian processes.}, number = {1}, urldate = {2023-06-06}, journal = {The Annals of Probability}, author = {Csorgo, Sandor}, month = feb, year = {1981}, note = {Publisher: Institute of Mathematical Statistics}, keywords = {60E05, 60F05, 60F15, 60G17, 62G99, continuity of a Gaussian process, Convergence rates, Empirical characteristic process, Fernique, Fernique inequality, Fernique-Marcus-Shepp theorem, Jain and Marcus, Komlos-Major-Tusnady theorem, stochastic integral, Strassen-type log log law, strong approximation, theorems of Dudley, weak convergence}, pages = {130--144}, file = {Full Text PDF:/Users/soumikp/Zotero/storage/NKFF2EVH/Csorgo - 1981 - Limit Behaviour of the Empirical Characteristic Fu.pdf:application/pdf}, }

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