統計数学セミナー

過去の記録 ~03/28次回の予定今後の予定 03/29~

担当者 吉田朋広、荻原哲平、小池祐太
セミナーURL http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/
目的 確率統計学およびその関連領域に関する研究発表, 研究紹介を行う.

2023年03月08日(水)

14:00-   数理科学研究科棟(駒場) 号室
Zoomによるハイブリッド配信(3/6申込締切)と現地参加(東京大学本郷キャンパス)          https://docs.google.com/forms/d/e/1FAIpQLSckefFqzVsTMDOr-5u1JN1_P8gNA7oZduP0QfTSP-OZ-w3qJQ/viewform
Evgeny Spodarev 氏 ( Ulm University, Germany)
Non-ergodic statistics for hamonizable stable processes (English)
[ 講演概要 ]
We consider stationary real harmonizable symmetric α-stable processes X={X(t):t∈ℝ} with a finite control measure. Assuming the control measure is symmetric and absolutely continuous with respect to the Lebesgue measure on the real line, we refer to its density function as the spectral density of X. Standard methods for statistical inference on stable processes cannot be applied as harmonizable stable processes are non-ergodic.
A stationary real harmonizable symmetric α-stable process X admits a LePage series representation and is conditionally Gaussian which allows us to derive the non-ergodic limit of sample functions on X. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of X and the lag process {X(t+h)−X(t):t∈ℝ} with h>0, respectively.
The process admits an equivalent representation as a series of sinusoidal waves with random frequencies whose probability density function is in fact the (normalized) spectral density of X. Using the strongly consistent frequency estimation via periodograms we present a strongly consistent estimator of the spectral density which is based only on one sampled path of X. The periodogram computation is fast and efficient, and our method is not affected by the non-ergodicity of X. Most notably no prior knowledge on parameters of the process such as its index of stability α is needed.

References:
[1] L.V. Hoang, E. Spodarev, "Inversion of alpha-sine and alpha-cosine transforms on R", Inverse Problems 37 (2021), 085008
[2] L.V. Hoang, E. Spodarev, "Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes", Preprint arXiv:2209.04315, submitted, 2022.
[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSet6w12XsqdCGQ8yEe4sOqRlCOhhrJXeKl5H7lMaRy4LZhmqQ/viewform