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This paper mainly addresses the optimization of $p$-th moment of$\mathbb{R}^n$-valued random variable. Through an ingenious approximationmechanism, one transforms the maximization problem into a sequence ofminimization problems, which can be converted into a sequence of nonlineardifferential equations with constraints by variational approach. The existenceand uniqueness of the solution for each equation can be demonstrated byapplying the canonical duality method. Moreover, the dual transformation givesa sequence of perfect dual maximization problems. In the final analysis, oneconstructs the approximation of the probability density function accordingly.

An approximation method for the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable