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The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative <i>m</i>-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain asymptotic expressions for the moments of random ratios. It is pointed out that our random ratios can include some models such as change-point detection. Last, some simulations are illustrated to test our results.