Contributions to the theory of weak dependence [microform] / by Brendan K. Beare
- Bib ID:
- 4498819
- Format:
- Book and Microform
- Author:
- Beare, Brendan K
- Online Version:
- http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3293289
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- Description:
- 92 p.
- ISBN:
- 9780549373421
- Summary:
-
This dissertation consists of three chapters, each of which is concerned with the notion of weak dependence between groups of random variables. The three chapters are respectively titled A Generalization of Hoffding's Lemma, A New Characterization of Mixing, and Markov Chains, Copulas, and beta-mixing. In A Generalization of Hoffding's Lemma, I extend the well known lemma of Hoffding (1940) to a multivariate setting. Hoffding's Lemma states that, for random variables X and Y with joint distribution FX,Y and marginal distributions FX and FY, we have CovX,Y =-infinityinfinity -infinityinfinityF X,Yx,y -FXx FYy dxdy, provided that E[divides]X[divides] < infinity, E[divides]Y[divides] < infinity and E[divides]XY[divides] < infinity. I show that this result can be generalized to apply to covariances between smooth real valued functions of several random variables. The multivariate version of Hoffding's Lemma leads naturally to a new class of covariance inequalities.
In particular, I prove that if the dependence between random vectors X and Y can be bounded in a certain sense by some gamma < infinity, then for smooth functions f and g, we have CovfX ,gY <=g fHK gHK. The norm [parallel]
I term this form of mixing gamma-mixing. Like alpha-mixing, gamma-mixing is defined in terms of distances of the form [divides]P( A[intersection]B - P(A) P(B)[divides], where (loosely speaking) A is an event depending on random variables in the distant past, while B is an event depending on random variables in the distant future. However, whereas alpha-mixing coefficients are constructed by taking the supremum of this distance over entire sigma-fields of sets, I define gamma-mixing coefficients by taking the supremum over a smaller class of sets: the finite dimensional cylinder sets. This leads to a definition of mixing more general than existing characterizations. The generalization of Hoffding's Lemma proved in Chapter 1 is used to prove covariance inequalities for functions of gamma-mixing processes.
These covariance inequalities are used to establish weak and strong laws of large numbers, a Rosenthal inequality, and a functional central limit theorem, all of which apply to time series exhibiting a suitable tradeoff between the existence of higher order moments and the rate of decay of gamma-mixing coefficients. In the third chapter, Markov Chains, Copulas, and beta-mixing, I consider the relationship between the finite dimensional copulas of a stationary Markov chain, and the property of beta-mixing. More particularly, I prove the following result: if the copula of consecutive observations in a stationary Markov chain is absolutely continuous with square integrable density, and has maximal correlation coefficient rho C less than one, then the Markov chain is beta-mixing with an exponentially fast rate of decay of mixing coefficients.
I show further that all absolutely continuous copulas exhibiting positive tail dependence have non-square integrable densities, while the condition rho C < 1 implies that the Markov chain is aperiodic. These results are relevant to recent research in econometrics in which Markov chains are modeled using a copula-based approach.
- Notes:
-
- (UMI)AAI3293289
- Source: Dissertation Abstracts International, Volume: 68-12, Section: B, page: 8108.
- Adviser: Peter C. B. Phillips.
- Thesis (Ph.D.)--Yale University, 2007.
- Reproduction:
- Microfilm. Ann Arbor, Mich. : University Microfilms International.
- Subject:
- Statistics
- Other authors/contributors:
- Yale University
- Copyright:
-
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Copyright status was determined using the following information:
- Material type:
- Literary Dramatic Musical
- Published status:
- Unpublished
- Creation date:
- 2007
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