We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability, classification of random sets by likelihood ratio tests, decomposition of complexity and independence, Bayesian statistics for individual random sequences. Formerly Lambalgen's theorem for correlated probability is shown under a uniform computability assumption in [H. Takahashi Inform. Comp. 2008]. In this paper we show the theorem without the assumption.