In this paper, we obtain the Hájek-Rényi-type inequality for a class of
random variable sequences and give some applications for associated random variable
sequences, strongly positive dependent stochastic sequences and martingale difference
sequences which generalize and improve the results of Prakasa Rao and Soo published
in Statist. Probab. Lett., 57(2002) and 78(2008). Using this result, we get the integrability of supremum and the strong law of large numbers for a class of random
variable sequences.
