In this paper we generalize the Hájek-Rényi-Chow maximal inequality for submartingales to L p type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for 1 ≤ p ≤ 2.

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T , in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional expectation operators, Positivity, 19 (2015), 647-657] as T -strong convergence and convergence in Tconditional probability, respectively. Generalized L p spaces for the cases of p = 1, 2, ∞, were discussed in the setting of Riesz spaces as L p (T ) spaces in [C. C. A. Labuschagne, B. A. Watson, Discrete stochastic integration in Riesz spaces, Positivity, 14 (2010), 859-875]. An R(T ) valued norm, for the cases of p = 1, ∞, was introduced on these spaces in [W. Kuo, M. Rogans, B.A. Watson, Mixing processes in Riesz spaces, Journal of Mathematical Analysis and Application, 456 (2017), 992-1004] where it was also shown that R(T ) is a universally complete f -algebra and that these spaces are R(T )-modules.

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T , in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional expectation operators, Positivity, 19 (2015), 647-657] as T -strong convergence and convergence in Tconditional probability, respectively. Generalized L p spaces for the cases of p = 1, 2, ∞, were discussed in the setting of Riesz spaces as L p (T ) spaces in [C. C. A. Labuschagne, B. A. Watson, Discrete stochastic integration in Riesz spaces, Positivity, 14 (2010), 859-875]. An R(T ) valued norm, for the cases of p = 1, ∞, was introduced on these spaces in [W. Kuo, M. Rogans, B.A. Watson, Mixing processes in Riesz spaces, Journal of Mathematical Analysis and Application, 456 (2017), 992-1004] where it was also shown that R(T ) is a universally complete f -algebra and that these spaces are R(T )-modules.

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