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Seminar on Probability and Statistics
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Organizer(s) | Nakahiro Yoshida, Hiroki Masuda, Teppei Ogihara, Yuta Koike |
|---|
Seminar information archive
2025/04/24
10:00-11:10 Room #126 (Graduate School of Math. Sci. Bldg.)
ハイブリッド開催
Stefano M. Iacus (Harvard University)
Inference for Ergodic Network Stochastic Differential Equations (English)
https://u-tokyo-ac-jp.zoom.us/meeting/register/cx7BR8oJSFGT42K4LY-fkQ
ハイブリッド開催
Stefano M. Iacus (Harvard University)
Inference for Ergodic Network Stochastic Differential Equations (English)
[ Abstract ]
We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (momentum effect), feedback from neighboring nodes (network effect), and a "stochastic volatility” term modeled by Brownian motion.
Our objective is to estimate the parameters of the N-SDE system under two different schemas: high-frequency discrete-time observations and small noise continuous-time observations.
The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph.
We consider two distinct scenarios: i) known network structure: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the quadratic growth of the parameter space with the number of edges. ii) unknown network structure: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models.
In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems.
Through simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.
[ Reference URL ]We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (momentum effect), feedback from neighboring nodes (network effect), and a "stochastic volatility” term modeled by Brownian motion.
Our objective is to estimate the parameters of the N-SDE system under two different schemas: high-frequency discrete-time observations and small noise continuous-time observations.
The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph.
We consider two distinct scenarios: i) known network structure: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the quadratic growth of the parameter space with the number of edges. ii) unknown network structure: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models.
In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems.
Through simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.
https://u-tokyo-ac-jp.zoom.us/meeting/register/cx7BR8oJSFGT42K4LY-fkQ
2025/02/20
13:00-14:10 Room #056 (Graduate School of Math. Sci. Bldg.)
Evgeny Spodarev (Universität Ulm)
Non-ergodic statistics for stationary-increment harmonizable stable processes (English)
https://docs.google.com/forms/d/e/1FAIpQLSd5_4NM3xazVUAARMhyv_e55RsYZFyfpOHqC0oGYasM2NLOqQ/viewform
Evgeny Spodarev (Universität Ulm)
Non-ergodic statistics for stationary-increment harmonizable stable processes (English)
[ Abstract ]
We consider the class of stationary-increment harmonizable stable processes $𝑋=\{ 𝑋(𝑡): 𝑡\in \mathbb{R} \}$ defined by $$𝑋(𝑡)=𝑅𝑒\left( \int_{\mathbb{R}} (𝑒^{𝑖𝑡𝑥}−1) \Psi (𝑥) 𝑀_{\alpha}(𝑑𝑥) \right), \quad 𝑡\in\mathbb{R},$$ where $𝑀_{\alpha}$ is an isotropic complex symmetric $\alpha$-stable (𝑆$\alpha$𝑆) random measure with Lebesgue control measure. This class contains real harmonizable fractional stable motions, which are a generalization of the harmonizable representation of fractional Brownian motions to the stable regime, when $\Psi(𝑥)=|𝑥|−𝐻−1/\alpha, 𝑥\in\mathbb{R}$. We give conditions for the integrability of the path of $𝑋$ with respect to a finite, absolutely continuous measure, and show that the convolution with a suitable measure yields a real stationary harmonizable 𝑆$\alpha$𝑆 process with finite control measure. Such a process admits a LePage type series representation consisting of sine and cosine functions with random amplitudes and frequencies, which can be estimated consistently using the periodogram. Combined with kernel density estimation, this allows us to construct consistent estimators for the index of stability $\alpha$ as well as the kernel function $\Psi$ in the integral representation of $𝑋$ (up to a constant factor). For real harmonizable fractional stable motions consistent estimators for the index of stability $\alpha$ and its Hurst parameter $𝐻$ are given, which are computed directly from the periodogram frequency estimates.
[ Reference URL ]We consider the class of stationary-increment harmonizable stable processes $𝑋=\{ 𝑋(𝑡): 𝑡\in \mathbb{R} \}$ defined by $$𝑋(𝑡)=𝑅𝑒\left( \int_{\mathbb{R}} (𝑒^{𝑖𝑡𝑥}−1) \Psi (𝑥) 𝑀_{\alpha}(𝑑𝑥) \right), \quad 𝑡\in\mathbb{R},$$ where $𝑀_{\alpha}$ is an isotropic complex symmetric $\alpha$-stable (𝑆$\alpha$𝑆) random measure with Lebesgue control measure. This class contains real harmonizable fractional stable motions, which are a generalization of the harmonizable representation of fractional Brownian motions to the stable regime, when $\Psi(𝑥)=|𝑥|−𝐻−1/\alpha, 𝑥\in\mathbb{R}$. We give conditions for the integrability of the path of $𝑋$ with respect to a finite, absolutely continuous measure, and show that the convolution with a suitable measure yields a real stationary harmonizable 𝑆$\alpha$𝑆 process with finite control measure. Such a process admits a LePage type series representation consisting of sine and cosine functions with random amplitudes and frequencies, which can be estimated consistently using the periodogram. Combined with kernel density estimation, this allows us to construct consistent estimators for the index of stability $\alpha$ as well as the kernel function $\Psi$ in the integral representation of $𝑋$ (up to a constant factor). For real harmonizable fractional stable motions consistent estimators for the index of stability $\alpha$ and its Hurst parameter $𝐻$ are given, which are computed directly from the periodogram frequency estimates.
https://docs.google.com/forms/d/e/1FAIpQLSd5_4NM3xazVUAARMhyv_e55RsYZFyfpOHqC0oGYasM2NLOqQ/viewform
2025/02/07
14:00-15:10 Room #128 (Graduate School of Math. Sci. Bldg.)
ハイブリッド開催
Juho Leppänen (Tokai University)
A multivariate Berry–Esseen theorem for deterministic dynamical systems (English)
https://u-tokyo-ac-jp.zoom.us/meeting/register/lmbyLgO6RNi1GnoovqW_Sg
ハイブリッド開催
Juho Leppänen (Tokai University)
A multivariate Berry–Esseen theorem for deterministic dynamical systems (English)
[ Abstract ]
Many chaotic deterministic dynamical systems with a random initial state satisfy limit theorems similar to those of independent random variables. A classical example is the Central Limit Theorem, which, for a broad class of ergodic measure-preserving systems, is known to follow from a sufficiently rapid decay of correlations. Much work has also been done on the rate of convergence in the CLT. Results in this area typically rely on additional structure, such as suitable martingale approximation schemes or a spectral gap for the Perron–Frobenius operator.
In this talk, we present an adaptation of Stein's method for multivariate normal approximation of deterministic dynamical systems. For vector-valued processes generated by a class of fibred systems with good distortion properties (Gibbs–Markov maps), we derive bounds on the convex distance between the distribution of scaled partial sums and a multivariate normal distribution. These bounds, which are deduced as a consequence of certain correlation decay criteria, involve a multiplicative constant whose dependence on the dimension and dynamical quantities is explicit.
[ Reference URL ]Many chaotic deterministic dynamical systems with a random initial state satisfy limit theorems similar to those of independent random variables. A classical example is the Central Limit Theorem, which, for a broad class of ergodic measure-preserving systems, is known to follow from a sufficiently rapid decay of correlations. Much work has also been done on the rate of convergence in the CLT. Results in this area typically rely on additional structure, such as suitable martingale approximation schemes or a spectral gap for the Perron–Frobenius operator.
In this talk, we present an adaptation of Stein's method for multivariate normal approximation of deterministic dynamical systems. For vector-valued processes generated by a class of fibred systems with good distortion properties (Gibbs–Markov maps), we derive bounds on the convex distance between the distribution of scaled partial sums and a multivariate normal distribution. These bounds, which are deduced as a consequence of certain correlation decay criteria, involve a multiplicative constant whose dependence on the dimension and dynamical quantities is explicit.
https://u-tokyo-ac-jp.zoom.us/meeting/register/lmbyLgO6RNi1GnoovqW_Sg
2024/07/23
15:00-16:10 Room #118 (Graduate School of Math. Sci. Bldg.)
田栗 正隆 (東京医科大学医療データサイエンス分野)
近似的な多重頑健推定量を用いた時間依存性交絡の調整 (日本語)
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZcocOGgrDIpHtIPBLecsHgqaY6tjuNB4Voc
田栗 正隆 (東京医科大学医療データサイエンス分野)
近似的な多重頑健推定量を用いた時間依存性交絡の調整 (日本語)
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZcocOGgrDIpHtIPBLecsHgqaY6tjuNB4Voc
2024/06/28
13:00-14:10 Room #128 (Graduate School of Math. Sci. Bldg.)
原田 和治 (東京医科大学医療データサイエンス分野)
医学における予測モデルの活用と階層構造を持つ順序回帰の提案 (日本語)
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUpd-ispjIqG9NfJk7_kjW2pBcvq_KMXHPW
原田 和治 (東京医科大学医療データサイエンス分野)
医学における予測モデルの活用と階層構造を持つ順序回帰の提案 (日本語)
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUpd-ispjIqG9NfJk7_kjW2pBcvq_KMXHPW
2024/06/18
13:00-14:10 Room #128 (Graduate School of Math. Sci. Bldg.)
Lorenzo Mercuri (University of Milan)
A compound CARMA(p,q)-Hawkes process for pricing financial derivatives (English)
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZ0rcOmvpjwuGNHx8ht0rMs1rD3HcEajoJv6
Lorenzo Mercuri (University of Milan)
A compound CARMA(p,q)-Hawkes process for pricing financial derivatives (English)
[ Abstract ]
Recently, a new self-exciting point process with a continuous-time autoregressive moving average intensity process, named CARMA(p,q)-Hawkes model, has been introduced. The model generalizes the well-known Hawkes process by substituting the Ornstein-Uhlenbeck intensity with a CARMA(p,q) model where the associated state process is driven by the counting process itself. The new model maintains the same level of tractability of the Hawkes (e.g., Infinitesimal generator, backward and forward Kolmogorov equation, joint characteristic function and so on). However, it is able to reproduce more complex time-dependency structure observed in several market data.
Starting from this model, we introduce a Compound CARMA(p,q)-Hawkes with a random jump size independent of the counting and the intensity processes. This can be used as the main block for a new option pricing model, due to log-affine structure of the characteristic function of the underlying log-price driven by a pure jump compound CARMA(p,q)-Hawkes.
Further, we extend this model by scaling it with a measurable function of the time and the left-limit of the price itself. Exploiting the Markov structure of the new model, we derive the forward Kolmogorov equation that leads us to a Dupire-like formula. Some numerical results will also be presented.
[ Reference URL ]Recently, a new self-exciting point process with a continuous-time autoregressive moving average intensity process, named CARMA(p,q)-Hawkes model, has been introduced. The model generalizes the well-known Hawkes process by substituting the Ornstein-Uhlenbeck intensity with a CARMA(p,q) model where the associated state process is driven by the counting process itself. The new model maintains the same level of tractability of the Hawkes (e.g., Infinitesimal generator, backward and forward Kolmogorov equation, joint characteristic function and so on). However, it is able to reproduce more complex time-dependency structure observed in several market data.
Starting from this model, we introduce a Compound CARMA(p,q)-Hawkes with a random jump size independent of the counting and the intensity processes. This can be used as the main block for a new option pricing model, due to log-affine structure of the characteristic function of the underlying log-price driven by a pure jump compound CARMA(p,q)-Hawkes.
Further, we extend this model by scaling it with a measurable function of the time and the left-limit of the price itself. Exploiting the Markov structure of the new model, we derive the forward Kolmogorov equation that leads us to a Dupire-like formula. Some numerical results will also be presented.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZ0rcOmvpjwuGNHx8ht0rMs1rD3HcEajoJv6
2024/04/10
13:30-14:40 Room #126 (Graduate School of Math. Sci. Bldg.)
Ivan Nourdin (University of Luxembourg)
Limit theorems for additive functionals of stationary Gaussian fields (English)
https://forms.gle/uMKm3gVquLpYaVdc6
Ivan Nourdin (University of Luxembourg)
Limit theorems for additive functionals of stationary Gaussian fields (English)
[ Abstract ]
In this talk, we will investigate central and non-central limit theorems for additive functionals of stationary Gaussian fields. Our main tool will be the Malliavin-Stein approach. Based on joint works with Nikolai Leonenko, Leonardo Maini and Francesca Pistolato.
[ Reference URL ]In this talk, we will investigate central and non-central limit theorems for additive functionals of stationary Gaussian fields. Our main tool will be the Malliavin-Stein approach. Based on joint works with Nikolai Leonenko, Leonardo Maini and Francesca Pistolato.
https://forms.gle/uMKm3gVquLpYaVdc6
2023/03/08
14:00- Room # (Graduate School of Math. Sci. Bldg.)
Evgeny Spodarev ( Ulm University, Germany)
Non-ergodic statistics for hamonizable stable processes (English)
https://docs.google.com/forms/d/e/1FAIpQLSet6w12XsqdCGQ8yEe4sOqRlCOhhrJXeKl5H7lMaRy4LZhmqQ/viewform
Evgeny Spodarev ( Ulm University, Germany)
Non-ergodic statistics for hamonizable stable processes (English)
[ Abstract ]
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.
[ Reference URL ]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.
https://docs.google.com/forms/d/e/1FAIpQLSet6w12XsqdCGQ8yEe4sOqRlCOhhrJXeKl5H7lMaRy4LZhmqQ/viewform
2023/01/10
10:50-11:30 Room # (Graduate School of Math. Sci. Bldg.)
井口優雅 (University College London)
Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions
(現地参加) https://forms.gle/qwssLccVgsAWcfps7 (Zoom参加) (1/8迄) https://docs.google.com/forms/d/e/1FAIpQLSe7OYeMDfaZ7pTLO42k43Tn5dWKpsyw
井口優雅 (University College London)
Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions
[ Abstract ]
Parametric inference for multi-dimensional diffusion processes has been studied over the past decades. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) over finite time periods. Especially in the setting of some class of hypo-elliptic models, recent research (Ditlevsen and Samson 2019, Gloter and Yoshida 2021) has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates. In our work, first, we develop two weak second order ‘sampling schemes' (to cover both the hypo-elliptic and elliptic classes) and generate accompanying ‘transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods, in both high and low frequency observations regime. We also present numerical results from carrying out classical or Bayesian inference. This is a joint work with Alexandros Beskos and Matthew Graham, and the preprint is available at https://arxiv.org/abs/2211.16384.
[ Reference URL ]Parametric inference for multi-dimensional diffusion processes has been studied over the past decades. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) over finite time periods. Especially in the setting of some class of hypo-elliptic models, recent research (Ditlevsen and Samson 2019, Gloter and Yoshida 2021) has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates. In our work, first, we develop two weak second order ‘sampling schemes' (to cover both the hypo-elliptic and elliptic classes) and generate accompanying ‘transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods, in both high and low frequency observations regime. We also present numerical results from carrying out classical or Bayesian inference. This is a joint work with Alexandros Beskos and Matthew Graham, and the preprint is available at https://arxiv.org/abs/2211.16384.
(現地参加) https://forms.gle/qwssLccVgsAWcfps7 (Zoom参加) (1/8迄) https://docs.google.com/forms/d/e/1FAIpQLSe7OYeMDfaZ7pTLO42k43Tn5dWKpsyw
2022/12/05
14:40-15:50 Room # (Graduate School of Math. Sci. Bldg.)
Michael Choi (National University of Singapore and Yale-NUS College)
A binary branching model with Moran-type interactions (English)
(Zoom参加) 12/1締切https://docs.google.com/forms/d/e/1FAIpQLSdyluSozvNOGmDcXzGv496v2AQNiPePqIerLaBN9pD4wxEmnw/viewform (現地参加) 先着20名https://forms.gle/rS9rjhL2jXo6eGgt5
Michael Choi (National University of Singapore and Yale-NUS College)
A binary branching model with Moran-type interactions (English)
[ Abstract ]
Branching processes naturally arise as pertinent models in a variety of applications such as population size dynamics, neutron transport and cell proliferation kinetics. A key result for understanding the behaviour of such systems is the Perron Frobenius decomposition, which allows one to characterise the large time average behaviour of the branching process via its leading eigenvalue and corresponding left and right eigenfunctions. However, obtaining estimates of these quantities can be challenging, for example when the branching process is spatially dependent with inhomogeneous rates. In this talk, I will introduce a new interacting particle model that combines the natural branching behaviour of the underlying process with a selection and resampling mechanism, which allows one to maintain some control over the system and more efficiently estimate the eigenelements. I will then present the main result, which provides an explicit relation between the particle system and the branching process via a many-to-one formula and also quantifies the L^2 distance between the occupation measures of the two systems. Finally, I will discuss some examples in order to illustrate the scope and possible extensions of the model, and to provide some comparisons with the Fleming Viot interacting particle system. This is based on work with Alex Cox (University of Bath) and Denis Villemonais (Université de Lorraine).
[ Reference URL ]Branching processes naturally arise as pertinent models in a variety of applications such as population size dynamics, neutron transport and cell proliferation kinetics. A key result for understanding the behaviour of such systems is the Perron Frobenius decomposition, which allows one to characterise the large time average behaviour of the branching process via its leading eigenvalue and corresponding left and right eigenfunctions. However, obtaining estimates of these quantities can be challenging, for example when the branching process is spatially dependent with inhomogeneous rates. In this talk, I will introduce a new interacting particle model that combines the natural branching behaviour of the underlying process with a selection and resampling mechanism, which allows one to maintain some control over the system and more efficiently estimate the eigenelements. I will then present the main result, which provides an explicit relation between the particle system and the branching process via a many-to-one formula and also quantifies the L^2 distance between the occupation measures of the two systems. Finally, I will discuss some examples in order to illustrate the scope and possible extensions of the model, and to provide some comparisons with the Fleming Viot interacting particle system. This is based on work with Alex Cox (University of Bath) and Denis Villemonais (Université de Lorraine).
(Zoom参加) 12/1締切https://docs.google.com/forms/d/e/1FAIpQLSdyluSozvNOGmDcXzGv496v2AQNiPePqIerLaBN9pD4wxEmnw/viewform (現地参加) 先着20名https://forms.gle/rS9rjhL2jXo6eGgt5
2022/10/21
①14:30-15:40- ②16:20-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Estate Khmaladze (Victoria University of Wellington)
On the theory of distribution free testing of statistical hypothesis
①Empirical processes for discrete and continuous observations: structure, difficulties and resolution.
②Further testing problems: parametric regression and Markov chains. (ENGLISH)
https://docs.google.com/forms/d/e/1FAIpQLScxh_wNRs3WbMUG4S3cGlGAu1ZkP4trLbc08CBrvUDO66hwNg/viewform?usp=sf_link
Estate Khmaladze (Victoria University of Wellington)
On the theory of distribution free testing of statistical hypothesis
①Empirical processes for discrete and continuous observations: structure, difficulties and resolution.
②Further testing problems: parametric regression and Markov chains. (ENGLISH)
[ Abstract ]
The concept of distribution free testing is familiar to all. Everybody, who heard about rank statistics, knows that the distribution of ranks is independent from the distribution of underlying random variables, provided this later is a continuous distribution on the real line. Everybody, who ever used classical goodness of fit tests like Kolmogorov - Smirnov test or Cram\'er-von Mises test, knows that the distribution of statistics of these tests is independent from the distribution of the underlying random variables, again, provided this distribution is a continuous distribution on the real line.
Development in subsequent decades revealed many cracks in existing theory and difficulties in extending the concept of distribution free testing to majority of interesting models. It gradually became clear that the new starting point is needed to expand the theory to these models.
In our lectures we first describe the current situation in empirical and related processes. Then we describe how the new approaches have been developed and what progress has been made.
Then we hope to show how the new approach can be naturally extended to the domain of stochastic processes, and how the important probabilistic models of the processes can be tested in distribution free way. In discrete time, results for Markov chains have been published in 2021. Extension to continuous time will be explored during the current visit to University of Tokyo.
[ Reference URL ]The concept of distribution free testing is familiar to all. Everybody, who heard about rank statistics, knows that the distribution of ranks is independent from the distribution of underlying random variables, provided this later is a continuous distribution on the real line. Everybody, who ever used classical goodness of fit tests like Kolmogorov - Smirnov test or Cram\'er-von Mises test, knows that the distribution of statistics of these tests is independent from the distribution of the underlying random variables, again, provided this distribution is a continuous distribution on the real line.
Development in subsequent decades revealed many cracks in existing theory and difficulties in extending the concept of distribution free testing to majority of interesting models. It gradually became clear that the new starting point is needed to expand the theory to these models.
In our lectures we first describe the current situation in empirical and related processes. Then we describe how the new approaches have been developed and what progress has been made.
Then we hope to show how the new approach can be naturally extended to the domain of stochastic processes, and how the important probabilistic models of the processes can be tested in distribution free way. In discrete time, results for Markov chains have been published in 2021. Extension to continuous time will be explored during the current visit to University of Tokyo.
https://docs.google.com/forms/d/e/1FAIpQLScxh_wNRs3WbMUG4S3cGlGAu1ZkP4trLbc08CBrvUDO66hwNg/viewform?usp=sf_link
2022/10/19
10:30-11:40 Room # (Graduate School of Math. Sci. Bldg.)
Hayate Yamagishi (Graduate School of Mathematical Sciences, The University of Tokyo)
https://docs.google.com/forms/d/e/1FAIpQLSd3i_gFci4Dc8T8gjtMigm08aIoQH6gM_Yfw0bHfppM1CNmag/viewform?usp=sf_link
Hayate Yamagishi (Graduate School of Mathematical Sciences, The University of Tokyo)
[ Abstract ]
[ Reference URL ]https://docs.google.com/forms/d/e/1FAIpQLSd3i_gFci4Dc8T8gjtMigm08aIoQH6gM_Yfw0bHfppM1CNmag/viewform?usp=sf_link
2022/07/21
13:30-14:40 Room #- (Graduate School of Math. Sci. Bldg.)
( )
[ Reference URL ]
https://forms.gle/JrtVRcQNgn9pug3F7
( )
[ Reference URL ]
https://forms.gle/JrtVRcQNgn9pug3F7
2022/02/16
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Teppei Ogihara (University of Tokyo)
Efficient estimation for ergodic jump-diffusion processes
https://docs.google.com/forms/d/e/1FAIpQLSeRTEo19DJgFiVsEpLrRapqzkL6LZAiUMGdA0ezK-nWYSPrGg/viewform
Teppei Ogihara (University of Tokyo)
Efficient estimation for ergodic jump-diffusion processes
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
We study the estimation problem of the parametric model for ergodic jump-diffusion processes. Shimizu and Yoshida (Stat. Inference Stoch. Process. 2006) proposed a quasi-maximum-likelihood estimator based on a thresholding likelihood function that detects the existence of jumps.
In this talk, we consider the efficiency of estimators by using local asymptotic normality (LAN). To show the LAN property, we need to specify the asymptotic behavior of log-likelihood ratios, which is complicated for the jump-diffusion model because the transition probability for no jump is quite different from that for the presence of jumps. We develop techniques to show the LAN property based on transition density approximation. By applying these techniques to the thresholding likelihood function, we obtain the LAN property for the jump-diffusion model. Moreover, we have the asymptotic efficiency of
the quasi-maximum-likelihood estimator in Shimizu and Yoshida (2006) and a Bayes-type estimator proposed in Ogihara and Yoshida (Stat.Inference Stoch. Process. 2011). This is a joint work with Yuma Uehara (Kansai University).
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
We study the estimation problem of the parametric model for ergodic jump-diffusion processes. Shimizu and Yoshida (Stat. Inference Stoch. Process. 2006) proposed a quasi-maximum-likelihood estimator based on a thresholding likelihood function that detects the existence of jumps.
In this talk, we consider the efficiency of estimators by using local asymptotic normality (LAN). To show the LAN property, we need to specify the asymptotic behavior of log-likelihood ratios, which is complicated for the jump-diffusion model because the transition probability for no jump is quite different from that for the presence of jumps. We develop techniques to show the LAN property based on transition density approximation. By applying these techniques to the thresholding likelihood function, we obtain the LAN property for the jump-diffusion model. Moreover, we have the asymptotic efficiency of
the quasi-maximum-likelihood estimator in Shimizu and Yoshida (2006) and a Bayes-type estimator proposed in Ogihara and Yoshida (Stat.Inference Stoch. Process. 2011). This is a joint work with Yuma Uehara (Kansai University).
https://docs.google.com/forms/d/e/1FAIpQLSeRTEo19DJgFiVsEpLrRapqzkL6LZAiUMGdA0ezK-nWYSPrGg/viewform
2022/01/20
15:00-16:10 Room # (Graduate School of Math. Sci. Bldg.)
Yoshimasa Uematsu (-)
-
[ Reference URL ]
https://docs.google.com/forms/d/e/1FAIpQLSdH8oP72k7-qsHigZBBZ4F6N-bGIJ6BcOWgKLhted2ohGSBeg/viewform
Yoshimasa Uematsu (-)
-
[ Reference URL ]
https://docs.google.com/forms/d/e/1FAIpQLSdH8oP72k7-qsHigZBBZ4F6N-bGIJ6BcOWgKLhted2ohGSBeg/viewform
2022/01/19
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Martin Hazelton (Otago University)
Dynamic fibre samplers for linear inverse problems
https://docs.google.com/forms/d/e/1FAIpQLSeoXt5v8xdQNFAKTDLoD0lttaHjV17_r7864x11mtxU1EQlhQ/viewform
Martin Hazelton (Otago University)
Dynamic fibre samplers for linear inverse problems
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Statistical inverse problems occur when we wish to learn about some random process that is observed only indirectly. Inference in such situations typically involves sampling possible values for the latent variables of interest conditional on the indirect observations. For count data, the latent variables are constrained to lie on a fibre (solution set for the linear system) comprising the integer lattice within a convex polytope.
Sampling the latent counts can be conducted using MCMC methods,through a random walk on this fibre. A major challenge is finding a set of basic moves that ensures connectedness of the walk over the fibre. In principle this can be done by computing a Markov basis of potential moves, but the resulting sampler can be hugely inefficient even when such a basis is computable. In this talk I will describe some current work on developing a dynamic Markov basis that generates moves on the fly. This approach can guarantee irreducibility of the sampler while gaining efficiency by increasing the probability of selecting serviceable sampling directions.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Statistical inverse problems occur when we wish to learn about some random process that is observed only indirectly. Inference in such situations typically involves sampling possible values for the latent variables of interest conditional on the indirect observations. For count data, the latent variables are constrained to lie on a fibre (solution set for the linear system) comprising the integer lattice within a convex polytope.
Sampling the latent counts can be conducted using MCMC methods,through a random walk on this fibre. A major challenge is finding a set of basic moves that ensures connectedness of the walk over the fibre. In principle this can be done by computing a Markov basis of potential moves, but the resulting sampler can be hugely inefficient even when such a basis is computable. In this talk I will describe some current work on developing a dynamic Markov basis that generates moves on the fly. This approach can guarantee irreducibility of the sampler while gaining efficiency by increasing the probability of selecting serviceable sampling directions.
https://docs.google.com/forms/d/e/1FAIpQLSeoXt5v8xdQNFAKTDLoD0lttaHjV17_r7864x11mtxU1EQlhQ/viewform
2021/12/15
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Estate Khmaladze (Victoria University of Wellington)
Theory of Distribution-free Testing
https://docs.google.com/forms/d/e/1FAIpQLSdFj1XF8WJSPRmE0GFKY2QxscaGxC9msM6GkEsAf0TgD9yv2g/viewform
Estate Khmaladze (Victoria University of Wellington)
Theory of Distribution-free Testing
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
The aim of the talk is to introduce transformations of empirical-type processes by a group of unitary operators. Recall that if v_{nP} is empirical process on real line, based on a sample from P, it can be mapped into empirical process v_{nQ} by appropriate change of time
v_{nP}(h(x))=v_{nQ}(x)
where h(x) is continuous and increasing. This is the basis for distribution-free theory of goodness of fit testing. If w(\phi) is a function-parametric “empirical-type” process (i.e. has functions \phi from a space L as a time) and if K* is a unitary operator on L, then transformed process Kw we define as
Kw(\phi) = w(K*\phi)
These two formulas have good similarity, but one transformation in on the real line, while the other transformation in on functional space.This later one turns out to be of very broad use, and allows to base distribution-free theory upon it. Examples, we have specific results for, are parametric empirical
processes in R^d, regression empirical processes, those in GLM, parametric models for point processes and for Markov processes in discrete time. Hopefully, further examples will follow.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
The aim of the talk is to introduce transformations of empirical-type processes by a group of unitary operators. Recall that if v_{nP} is empirical process on real line, based on a sample from P, it can be mapped into empirical process v_{nQ} by appropriate change of time
v_{nP}(h(x))=v_{nQ}(x)
where h(x) is continuous and increasing. This is the basis for distribution-free theory of goodness of fit testing. If w(\phi) is a function-parametric “empirical-type” process (i.e. has functions \phi from a space L as a time) and if K* is a unitary operator on L, then transformed process Kw we define as
Kw(\phi) = w(K*\phi)
These two formulas have good similarity, but one transformation in on the real line, while the other transformation in on functional space.This later one turns out to be of very broad use, and allows to base distribution-free theory upon it. Examples, we have specific results for, are parametric empirical
processes in R^d, regression empirical processes, those in GLM, parametric models for point processes and for Markov processes in discrete time. Hopefully, further examples will follow.
https://docs.google.com/forms/d/e/1FAIpQLSdFj1XF8WJSPRmE0GFKY2QxscaGxC9msM6GkEsAf0TgD9yv2g/viewform
2021/11/17
15:30-17:00 Room # (Graduate School of Math. Sci. Bldg.)
Jean Bertoin (Institut of Mathematics, University of Zurich (UZH))
On the local times of noise reinforced Bessel processes
https://docs.google.com/forms/d/e/1FAIpQLSeuK9AOw6QUqvUge9ukw__v04j5jpfogzrGxlPLpEgNhW09kg/viewform
Jean Bertoin (Institut of Mathematics, University of Zurich (UZH))
On the local times of noise reinforced Bessel processes
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Bessel processes form a one-parameter family of self-similar diffusion on $[0,\infty)$ with the same Hurst exponent 1/2 as Brownian motion. Loosely speaking, in this setting, linear noise reinforcement with reinforcement parameter $p$ consists of repeating (if $p>0$) or counterbalancing (if $p<0$)infinitesimal increments of the process, uniformly at random and at a fixed rate as time passes. In this talk, we will investigate the effect of noise reinforcement on the local time at level $0$, that is, informally, the time that the process spends at $0$. A connection with increasing self-similar Markov processes will play a key role.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Bessel processes form a one-parameter family of self-similar diffusion on $[0,\infty)$ with the same Hurst exponent 1/2 as Brownian motion. Loosely speaking, in this setting, linear noise reinforcement with reinforcement parameter $p$ consists of repeating (if $p>0$) or counterbalancing (if $p<0$)infinitesimal increments of the process, uniformly at random and at a fixed rate as time passes. In this talk, we will investigate the effect of noise reinforcement on the local time at level $0$, that is, informally, the time that the process spends at $0$. A connection with increasing self-similar Markov processes will play a key role.
https://docs.google.com/forms/d/e/1FAIpQLSeuK9AOw6QUqvUge9ukw__v04j5jpfogzrGxlPLpEgNhW09kg/viewform
2021/10/13
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Li Cheng (National University of Singapore (NUS))
Bayesian Fixed-domain Asymptotics for Covariance Parameters in Gaussian Random Field Models
https://docs.google.com/forms/d/e/1FAIpQLSfEWrpkVavWEELx93dPxd0g2thhkC8NtA_8We4cDeiCKI6mZg/viewform
Li Cheng (National University of Singapore (NUS))
Bayesian Fixed-domain Asymptotics for Covariance Parameters in Gaussian Random Field Models
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Gaussian random field models are commonly used for modeling spatial processes. In this work we focus on the Gaussian process with isotropic Matern covariance functions. Under fixed-domain asymptotics,it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the range (or length-scale) parameter cannot. Motivated by this frequentist result, we prove that under a Bayesian fixed-domain framework, the posterior distribution of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior of the range parameter does not necessarily converge. Built on this new theory, we further show that the Bayesian kriging predictor satisfies the posterior asymptotic efficiency in linear prediction. We illustrate these asymptotic results in numerical examples.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Gaussian random field models are commonly used for modeling spatial processes. In this work we focus on the Gaussian process with isotropic Matern covariance functions. Under fixed-domain asymptotics,it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the range (or length-scale) parameter cannot. Motivated by this frequentist result, we prove that under a Bayesian fixed-domain framework, the posterior distribution of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior of the range parameter does not necessarily converge. Built on this new theory, we further show that the Bayesian kriging predictor satisfies the posterior asymptotic efficiency in linear prediction. We illustrate these asymptotic results in numerical examples.
https://docs.google.com/forms/d/e/1FAIpQLSfEWrpkVavWEELx93dPxd0g2thhkC8NtA_8We4cDeiCKI6mZg/viewform
2021/09/15
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Anup Biswas (Indian Institute of Science Education and Research (IISER), Pune)
Ergodic risk-sensitive control: history, new results and open problems
https://docs.google.com/forms/d/e/1FAIpQLSe-136jVBQwRDg3rgEGpgVtH2d4chXCvQuvnk_gE2fZqMGwBw/viewform
Anup Biswas (Indian Institute of Science Education and Research (IISER), Pune)
Ergodic risk-sensitive control: history, new results and open problems
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Risk-sensitive control became popular because of the robustness it provides to the optimal control. Its connection to the theory of large deviation also made it a natural candidate of mathematical interest. In this talk, we shall give an overview of the history of risk-sensitive control problems and some of its applications. We shall then (informally) discuss the ways of tackling this problem and the main questions of interest. At the end, we shall see some important open problems.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Risk-sensitive control became popular because of the robustness it provides to the optimal control. Its connection to the theory of large deviation also made it a natural candidate of mathematical interest. In this talk, we shall give an overview of the history of risk-sensitive control problems and some of its applications. We shall then (informally) discuss the ways of tackling this problem and the main questions of interest. At the end, we shall see some important open problems.
https://docs.google.com/forms/d/e/1FAIpQLSe-136jVBQwRDg3rgEGpgVtH2d4chXCvQuvnk_gE2fZqMGwBw/viewform
2021/08/18
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Gery Geenens (The University of New South Wales (UNSW Sydney))
Dependence, Sklar's copulas and discreteness
https://docs.google.com/forms/d/e/1FAIpQLScU9_QHdHZ-JeVyUIJOKUFmYJvG697NBDFkNh735WK9Cov1Og/viewform
Gery Geenens (The University of New South Wales (UNSW Sydney))
Dependence, Sklar's copulas and discreteness
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Yet the classical copula approach, building on Sklar’s theorem, cannot be legitimised if the variables of interest are not continuous. Indeed in the presence of discreteness, copula models are (i) unidentifiable, and (ii) not margin-free, and this by construction. In spite of the serious inconsistencies that this creates, downplaying statements are widespread in the literature, where copula methods are devised and used in discrete settings. In this work we call to reconsidering this current practice. To reconcile copulas with discreteness, we argued that they should be apprehended from a more fundamental perspective. Inspired by century-old ideas of Yule, we propose a novel construction which allows all the pleasant properties of copulas for modelling dependence (in particular:‘margin-freeness’) to smoothly carry over to the discrete setting.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Yet the classical copula approach, building on Sklar’s theorem, cannot be legitimised if the variables of interest are not continuous. Indeed in the presence of discreteness, copula models are (i) unidentifiable, and (ii) not margin-free, and this by construction. In spite of the serious inconsistencies that this creates, downplaying statements are widespread in the literature, where copula methods are devised and used in discrete settings. In this work we call to reconsidering this current practice. To reconcile copulas with discreteness, we argued that they should be apprehended from a more fundamental perspective. Inspired by century-old ideas of Yule, we propose a novel construction which allows all the pleasant properties of copulas for modelling dependence (in particular:‘margin-freeness’) to smoothly carry over to the discrete setting.
https://docs.google.com/forms/d/e/1FAIpQLScU9_QHdHZ-JeVyUIJOKUFmYJvG697NBDFkNh735WK9Cov1Og/viewform
2021/07/14
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Anirvan Chakraborty ( IISER Kolkata, India)
Statistics for Functional Data
https://docs.google.com/forms/d/e/1FAIpQLSfkHbmXT_3kHkBIUedzNSFqQ6QxuZzUQ9_qOgc8HqtZsKHTPQ/viewform
Anirvan Chakraborty ( IISER Kolkata, India)
Statistics for Functional Data
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
With the advancement in technology, statisticians often have to analyze data which are curves or functions observed over a domain. Data of this type is usually called functional data and is very common these days in various fields of science. Statistical modelling of this type of data is usually done by viewing the data as a random sample from a probability distribution on some infinite dimensional function space. This formulation, however, implies that one has to delve into the mathematical rigour and complexity of dealing with infinite dimensional objects and probability distributions in function spaces. As such, standard multivariate statistical methods are far from useful in analyzing such data. We will discuss some statistical techniques for analyzing functional data as well as outline some of the unique challenges faced and also discuss some interesting open problems in this frontline research area.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
With the advancement in technology, statisticians often have to analyze data which are curves or functions observed over a domain. Data of this type is usually called functional data and is very common these days in various fields of science. Statistical modelling of this type of data is usually done by viewing the data as a random sample from a probability distribution on some infinite dimensional function space. This formulation, however, implies that one has to delve into the mathematical rigour and complexity of dealing with infinite dimensional objects and probability distributions in function spaces. As such, standard multivariate statistical methods are far from useful in analyzing such data. We will discuss some statistical techniques for analyzing functional data as well as outline some of the unique challenges faced and also discuss some interesting open problems in this frontline research area.
https://docs.google.com/forms/d/e/1FAIpQLSfkHbmXT_3kHkBIUedzNSFqQ6QxuZzUQ9_qOgc8HqtZsKHTPQ/viewform
2021/07/03
10:55-17:10 Room # (Graduate School of Math. Sci. Bldg.)
- (-)
-
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/statmodel/?page_id=2028
- (-)
-
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/statmodel/?page_id=2028
2021/06/16
14:30-16:00 Room # (Graduate School of Math. Sci. Bldg.)
Hiroki Masuda (Kyushu University)
Levy-Ornstein-Uhlenbeck Regression
https://docs.google.com/forms/d/e/1FAIpQLSfkHbmXT_3kHkBIUedzNSFqQ6QxuZzUQ9_qOgc8HqtZsKHTPQ/viewform
Hiroki Masuda (Kyushu University)
Levy-Ornstein-Uhlenbeck Regression
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
We will present some of recent developments in parametric inference for a linear regression model driven by a non-Gaussian stable Levy process, when the process is observed at high frequency over a fixed time period. The model depends on a covariate process and the finite-dimensional parameter: the stability index (activity index) and the scale in the noise term, and the (auto)regression coefficients in the trend term, all being unknown. The maximum-likelihood estimator is shown to be asymptotically mixed-normally distributed with maximum concentration property. In order to bypass possible multiple-root problem and heavy numerical optimization, we also consider some easily computable initial estimator with which the one-step improvement does work. The asymptotic properties hold true in a unified manner regardless of whether the model is stationary and/or ergodic, almost without taking care of character of the
covariate process. Also discussed will be model-selection issues and some possible model extensions.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
We will present some of recent developments in parametric inference for a linear regression model driven by a non-Gaussian stable Levy process, when the process is observed at high frequency over a fixed time period. The model depends on a covariate process and the finite-dimensional parameter: the stability index (activity index) and the scale in the noise term, and the (auto)regression coefficients in the trend term, all being unknown. The maximum-likelihood estimator is shown to be asymptotically mixed-normally distributed with maximum concentration property. In order to bypass possible multiple-root problem and heavy numerical optimization, we also consider some easily computable initial estimator with which the one-step improvement does work. The asymptotic properties hold true in a unified manner regardless of whether the model is stationary and/or ergodic, almost without taking care of character of the
covariate process. Also discussed will be model-selection issues and some possible model extensions.
https://docs.google.com/forms/d/e/1FAIpQLSfkHbmXT_3kHkBIUedzNSFqQ6QxuZzUQ9_qOgc8HqtZsKHTPQ/viewform
2021/05/19
14:30-16:00 Online
Register at least 3 days before at the reference URL. The URL for participation sent before the seminar.
Federico Camia (NYU Abu Dhabi)
Limit Theorems and Random Fractal Curves in Statistical Mechanics (ENGLISH)
https://docs.google.com/forms/d/e/1FAIpQLSe2ObhY3dsFUUU4EaRyslLiAwfuA6chMmiw5uyfa1bvKMdyfg/viewform
Register at least 3 days before at the reference URL. The URL for participation sent before the seminar.
Federico Camia (NYU Abu Dhabi)
Limit Theorems and Random Fractal Curves in Statistical Mechanics (ENGLISH)
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Statistical mechanics deals with systems that have a large number of components whose behavior can often be considered random. For this reason, probability theory plays an essential role in the mathematical analysis of the subject. After a gentle introduction to statistical mechanics, I will give a nontechnical overview of recent, exciting developments that combine in a beautiful and unexpected way discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific models, percolation and the Ising model, which have a long history, have played an important part in the development of statistical mechanics, and occupy a central place in modern probability theory.
[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home
Statistical mechanics deals with systems that have a large number of components whose behavior can often be considered random. For this reason, probability theory plays an essential role in the mathematical analysis of the subject. After a gentle introduction to statistical mechanics, I will give a nontechnical overview of recent, exciting developments that combine in a beautiful and unexpected way discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific models, percolation and the Ising model, which have a long history, have played an important part in the development of statistical mechanics, and occupy a central place in modern probability theory.
https://docs.google.com/forms/d/e/1FAIpQLSe2ObhY3dsFUUU4EaRyslLiAwfuA6chMmiw5uyfa1bvKMdyfg/viewform
