Are asymptotic confidence sequences and anytime $p$-values uniformly valid for a nontrivial class of distributions $\mathcal{P}$? We give a positive answer to this question by deriving distribution-uniform anytime-valid inference procedures. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests that enable inference at stopping times -- have been justified nonasymptotically. Nevertheless, asymptotic procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and weak assumptions. While recent work has derived asymptotic analogues of anytime-valid methods with the aforementioned benefits, these were not shown to be $\mathcal{P}$-uniform, meaning that their asymptotics are not uniformly valid in a class of distributions $\mathcal{P}$. Indeed, the anytime-valid inference literature currently has no central limit theory to draw from that is both uniform in $\mathcal{P}$ and in the sample size $n$. This paper fills that gap by deriving a novel $\mathcal{P}$-uniform strong Gaussian approximation theorem, enabling $\mathcal{P}$-uniform anytime-valid inference for the first time. Along the way, our Gaussian approximation also yields a $\mathcal{P}$-uniform law of the iterated logarithm.

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