Decoupling From Dependence to Independence
Decoupling theory provides a general framework for analyzing problems involving dependent random variables as if they were independent. It was born in the early eighties as a natural continuation of martingale theory and has acquired a life of its own due
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The theory of decoupling aims at reducing the level of dependence in certain problems by means of inequalities that compare the original sequence to one involving independent random variables. It is therefore important to have information on results dealing with functionals of independent random variables. In this chapter we collect several types of results on sums of independent random variables that will be used throughout. We consider aspects of estimation of tail probabilities and moments that are relevant to the theory of decoupling and develop them to the extent needed, and, in a few instances, a little more. We begin with the classical Levy maximal inequalities, bounding the tail probabilities of the maximum of the norm of a sum of independent symmetric random vectors by the tail probabilities of the norm of the last sum, that is, the reflection principle for symmetric random walk extended to random variables taking values in a Banach space. Then, we also present analogous maximal inequalities for sums of arbitrary independent identically distributed random vectors. The proofs in the Banach space case are not more difficult than for the real case. A way to prove integrability for (classes of) random variables is to obtain bounds for tail probabilities in terms of the squares of these same probabilities at lower levels. This is illustrated by the Hoffmann-J\ZIrgensen type inequalities that we present in Section 2, which bound the pth moment of a sum of independent centered random vectors by a constant times the same moment of their maximum plus the pth power of a quantile. They are important as a means of upgrading
V. H. d e la Peña et al.,Decoupling © Springer Science+Business Media New York 1999
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Randomly Stopped Processes With Independent Increments
L-function from Section 1.5). In Section 2 we introduce good lambda inequalities that are used for developing Lp comparisons between correlated processes based on a special type of tail probability comparison, which is frequently manageable. They are used extensively in this chapter as well as in Chapters 6 and 7 in developing general decoupling and martingale inequalities. In Section 3 we present the main result of this chapter, a two-sided decoupling inequality for the Lp-norm of randomly stopped sums of independent random variables and give a proof of the right-hand side of this result. The proof of the left-hand side, being more complicated, is given in Section 4. Continuing in Section 5 we show how to extend the decoupling inequality to the case of continuous time processes with independent increments. The proof of this extension is based on limiting arguments. In the same section we also show how to apply this inequality to obtain sharp conditions for the validity ofWald's equation. These results can be interpreted as extensions ofWald's equations and support the heuristic that stopping times of processes with independent increments can (often) be viewed as random times independent of the associated processes. In Section 6, we develop an extens
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