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In this paper, we present an almost sure central limit theorem (ASCLT) forthe hyperbolic Anderson model (HAM) with a L\'evy white noise in afinite-variance setting. This work complements a recent work by the first andthird authors on the (quantitative) central limit theorems for the solution tothe HAM. We present two different proofs: one uses the Clark-Ocone formula andtakes advantage of the martingale structure of the white-in-time noise, and theother is obtained by combining the second-order Gaussian Poincar\'e inequalitywith Ibragimov and Lifshits' method of characteristic functions. Bothapproaches are different from the one developed in the PhD thesis of C. Zheng(2011), allowing us to establish the ASCLT without using lengthy computationsbased on star contractions. Moreover, the second approach is expected to beuseful for similar studies on SPDEs with Gaussian colored-in-time noise whenthe first approach, based on It\^o calculus, is not applicable.

Almost sure central limit theorem for the hyperbolic Anderson model with L\'evy white noise