Gnedenko–Raikov's theorem, central limit theory, and the weak law of large numbers

2006 ◽  
Vol 76 (17) ◽  
pp. 1935-1939 ◽  
Author(s):  
Allan Gut
Author(s):  
James Davidson

This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first provides a handbook and reference for the underlying mathematics (Part I, Chapters 1–6), statistical theory (Part II, Chapters 7–11), and stochastic process theory (Part III, Chapters 12–18). The second half provides a treatment of the main convergence theorems used in analysing the large sample behaviour of econometric estimators and tests. These are the law of large numbers (Part IV, Chapters 19–22), the central limit theorem (Part V, Chapters 23–26), and the functional central limit theorem (Part VI, Chapters 27–32). The focus in this treatment is on the nonparametric approach to time series properties, covering topics such as nonstationarity, mixing, martingales, and near‐epoch dependence. While the approach is not elementary, care is taken to keep the treatment self‐contained. Proofs are provided for almost all the results.


2018 ◽  
Vol 50 (A) ◽  
pp. 177-190
Author(s):  
Götz Kersting ◽  
Anton Wakolbinger

Abstract We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the logarithm of the block-counting process by means of a drifted subordinator.


2014 ◽  
Vol 51 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.


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