*James Davidson*

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0021
- Subject:
- Economics and Finance, Econometrics

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit ...
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This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.Less

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.

*Florence Merlevède, Magda Peligrad, and Sergey Utev*

- Published in print:
- 2019
- Published Online:
- April 2019
- ISBN:
- 9780198826941
- eISBN:
- 9780191865961
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198826941.003.0016
- Subject:
- Mathematics, Probability / Statistics

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling ...
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As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.Less

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or *ϕ*-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.