scholarly journals Limit theorems associated with the Pitman–Yor process

2017 ◽  
Vol 49 (2) ◽  
pp. 581-602
Author(s):  
Shui Feng ◽  
Fuqing Gao ◽  
Youzhou Zhou
Keyword(s):  
Disordered Systems ◽  
Limit Theorems ◽  
Large Deviation ◽  
Dirichlet Distribution ◽  
Discrete Measure ◽  
Random Energy Model ◽  
Large Deviation Principles ◽  
Large Numbers ◽  
Random Weights ◽  
Two Parameter

Abstract The Pitman–Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson–Dirichlet distribution with parameters 0 < α < 1, θ > -α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman–Yor process and the two-parameter Poisson–Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

scholarly journals Large deviations in non-uniformly hyperbolic dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 587-612 ◽  
Author(s):  
LUC REY-BELLET ◽  
LAI-SANG YOUNG
Keyword(s):  
Dynamical Systems ◽  
Generating Functions ◽  
Limit Theorems ◽  
Large Deviation ◽  
Return Times ◽  
Hyperbolic Diffeomorphisms ◽  
Large Deviation Principles ◽  
Hyperbolic Dynamical Systems ◽  
Rank One ◽  
Dispersing Billiards

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

scholarly journals Information Transmission under Random Emission Constraints

2014 ◽  
Vol 23 (6) ◽  
pp. 973-1009 ◽  
Author(s):  
FRANCIS COMETS ◽  
FRANÇOIS DELARUE ◽  
RENÉ SCHOTT
Keyword(s):  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Total Duration ◽  
Limited Resources ◽  
Large Numbers ◽  
Coupon Collector ◽  
Selection Of ◽  
Quantities Of Interest

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n → ∞, for the proportion of informed vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton–Watson tree with respect to the success epochs of the coupon collector problem.

scholarly journals Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

2014 ◽  
Vol 03 (03) ◽  
pp. 1450012 ◽  
Author(s):  
Jan Nagel
Keyword(s):  
Weak Convergence ◽  
Spectral Measure ◽  
Limit Theorems ◽  
Large Deviation ◽  
The Other ◽  
Limit Measure ◽  
Jacobi Ensemble ◽  
Large Deviation Principles ◽  
Beta Ensemble ◽  
Rate Functions

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

scholarly journals Large Number of Rare Events: Diversity Analysis in Multiple Choice Questionnaires and Related Topics

2021 ◽  
Author(s):  
◽  
Giorgi Kvizhinadze
Keyword(s):  
Statistical Theory ◽  
Limit Theorems ◽  
Edgeworth Expansion ◽  
Rare Events ◽  
Large Deviation ◽  
Diversity Analysis ◽  
Almost Everywhere ◽  
Large Numbers ◽  
The Subject ◽  
Opinion Pool

<p>The statistical analysis of a large number of rare events, (LNRE), which can also be called statistical theory of diversity, is the subject of acute interest both in statistical theory and in numerous applications. A careful eye will quickly see the presence of a large number of very rare objects almost everywhere: large numbers of rare species in ecosystems, large numbers of rare opinions in any opinion pool, large numbers of small admixtures in any solution and large numbers of rare words in any text are only few examples. In studying such objects, the interest for mathematical statisticians lies in the fact that most of the frequencies are small and, therefore, difficult to deal with. It is not immediately clear how one should be able to derive consistent and reliable inference from a large number of such frequencies. In this thesis we study the diversity of questionnaires with multiple answers. It has been demonstrated that this is a particular model of LNRE theory. In our analysis, the theories of large deviation, contiguity and Edgeworth expansion were employed, and limit theorems have been established.</p>

scholarly journals DYNAMIC QUANTUM BERNOULLI RANDOM WALKS

2008 ◽  
Vol 11 (02) ◽  
pp. 213-229
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
RENÉ SCHOTT
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Numbers

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (03) ◽  
pp. 693-710 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Position Distribution ◽  
Discrete State ◽  
Large Numbers ◽  
Watson Process ◽  
Positive Recurrent ◽  
Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (3) ◽  
pp. 693-710 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Position Distribution ◽  
Discrete State ◽  
Large Numbers ◽  
Watson Process ◽  
Positive Recurrent ◽  
Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanqing Wang ◽  
Quansheng Liu
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Moderate Deviation ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Branching Process With Immigration

Abstract This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 p\geq 1 , and the boundedness of the harmonic moments E ⁢ W n - a \mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log ⁡ Z n \log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.

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