Mixing inequalities in Riesz spaces

Abstract

Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the α (strong) and ϕ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized L 1 and L ∞ spaces begun by Kuo, Labuschagne and Watson will be extended. *

The Hájek-Rényi-Chow maximal inequality and a strong law of large numbers in Riesz spaces

“…The space E u is an f -algebra with multiplication defined so that the chosen weak order unit, e, is the algebraic unit. Further, it was shown in [16] that [19] and generalized to [3] where functional calculus was used to define f (x) = x p for x ∈ E u + . Much of the mathematical machinery needed to work in L p (T ), 1 < p < ∞, was developed in [10] even though such spaces were not considered there.…”

Near-epoch dependence in Riesz spaces

“…This is the case since 0 ≤ (f ± g) 2 = f 2 + g 2 ± 2f g, which gives 2|f g| ≤ f 2 + g 2 ∈ L 1 (T ), and so the claim follows from the fact that L 1 (T ) is an order ideal in E u . Recall from [11] that the L 1 (T ) and L ∞ (T ) spaces are R(T )-modules. The analogous result for L 2 (T ) is presented as follows.…”

“…From [11], we have that the T -conditional norms · T,1 : f → T |f | and · T,∞ : f → inf{g ∈ R(T ) + : |f | ≤ g} define R(T )-valued norms on L 1 (T ) and L ∞ (T ), respectively. To define an analogous T -conditional norm on L 2 (T ), we require the following result.…”

“…This extension was generalized to the Riesz space setting by Kuo, Labuschagne and Watson in [10] and enabled the definition of the generalized L p spaces in [1,11,14]. In particular, the natural domain of a Riesz space conditional expectation operator, T , is identified with the generalized L p space L 1 (T ) and T | · | defines an R(T ) + valued norm on L 1 (T ), see [11], where L ∞ (T ) and its R(T ) + valued norm are also defined. Building on this structure, using functional calculus, similar constructions can be made for p ∈ (1, ∞), see [1].…”

“…In [25] a Douglas-Andô-Radon-Nikodým theorem on Riesz spaces with a conditional expectation operator was proved. This gives the necessary tools for building and identifying conditional expectation operators on Riesz spaces, required in the study of Markov, [23,24] and mixing processes, [11]. The study of mixing processes in Riesz spaces began with the the formulation of mixingales in Riesz spaces in [12], where a law of large numbers was proved for Riesz space mixingales.…”