  In this paper we generalize the H\'ajek-R\'enyi-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 $1\le p\le 2$.
