| 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∣X∣ < infinity, E∣Y∣ < infinity and E∣XY∣ < 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 ∥·∥HK is the Hardy-Krause norm, a form of total variation for multivariate functions introduced in the early twentieth century which has received relatively little attention in more recent years. The inequality just stated is applied extensively in Chapter 2 to prove new covariance inequalities for time series that are weakly dependent in a certain sense.; In the second chapter, A New Characterization of Mixing, I propose a new definition of mixing, or asymptotic independence, for time series. I term this form of mixing gamma-mixing. Like alpha-mixing, gamma-mixing is defined in terms of distances of the form ∣P( A∩B - P(A) P(B)∣, 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. |
