  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. We apply some of these results to obtain
an anytime-valid test of conditional independence without the Model-X
assumption, as well as a $\mathcal{P}$-uniform law of the iterated logarithm.
