Some new Hardy inequalities in probability

Hardy et al. (1934) came up with Hardy's inequality in their book. Klaassen and Wellner (2021) gave the probability version of the Hardy inequality when the parameter p > 1. Based on their work, in this paper, we assign the randomness to variables as well. When p > 1, we give some extensions of Hardy's inequality. When 0 < p < 1, we provide the corresponding Hardy inequality in probability language. Also, we show that in some circumstances, our results contain the integral form of Hardy's inequality. We give a reversed Hardy inequality for random variables as well.