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This paper establishs the large deviation principle (LDP) for multipleaverages on $\mathbb{N}^d$. We extend the previous work of [Carinci et al.,Indag. Math. 2012] to multidimensional lattice $\mathbb{N}^d$ for $d\geq 2$.The same technique is also applicable to the weighted multiple average launchedby Fan [Fan, Adv. Math. 2021]. Finally, the boundary conditions are imposed tothe multiple sum and explicit formulae of the energy functions with respect tothe boundary conditions are obtained.

Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$