Let $\mathcal{X}\_n=\{ X\_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\Lambda$. Assume that for any $\lambda \in \Lambda$ it is known that $X\_{n,\lambda}\to 0$ almost surely, i.e. for any $\epsilon>0$
$$
P\{|X\_{n,\lambda}|>\epsilon \text{ i.o. }\}=0.
$$
Under which assumptions can we conclude that $P\{\sup\_{\lambda \in\Lambda}|X\_{n,\lambda}|>\epsilon \text{ i.o. }\}=0$ for all $\epsilon>0$?


I know that this problem is well studied in the case where $Y\_1, Y\_2, \ldots$ are independent and identically distributed random vectors, with common distribution $Q$, and each $X\_{n,\lambda}$ is of the form
$$
\frac{1}{n}\sum\_{i=1}^nf\_\lambda(Y\_i)-\int f\_\lambda(y)\text{d}Q(y),
$$
where $\mathcal{F}=\{f\_\lambda, \lambda \in \Lambda\}$ constitutes a class of measurable functions. In this context, appropriate metric entropy conditions yield a functional generalisation of the strong law of large numbers. Are you aware of results which hold true beyond the case of random variables in the form of centered means?



Almost sure convergence of the supremum over a class of random variables
The fundamental issue here is bounding the distribution of the supremum of a collection of random variables. The book "Upper and Lower Bounds for Stochastic Processes" by Michel Talagrand is largely devoted to this issue, <https://link.springer.com/book/10.1007/978-3-642-54075-2>
as is his earlier related book "the generic chaining".


