# Sum of infinite dimensional random variables

If $X_i$ are Independent and Identically Distributed (IID) vector valued variables with positive mean, and finite variance, then with Chebyshev's inequality, we know that their sum $S_n=\sum_{k\leq n} X_k$ will grow linearly with probability tending to 1, ie:

$$p(S_n \geq n (\mu - \epsilon) ) \rightarrow 1$$

where the $\geq$ is meant component wise

I'm trying to find conditions so that this is also true for infinite-dimensional random variables. That is, $f_i(x)$ are IID random functions of $I\rightarrow \mathbb{R}$ (where I is some finite interval $I=[a,b]$), with positive mean function $\mu(x)$, then find a strictly positive function B such that:

$$p(S_n \geq n B) \rightarrow 1$$

Where the $\geq$ means that at every point one function is bigger than the other

For example, I don't think this is true if the values at different x positions $f_i(x_0)$ and $f_i(x_1)$ are completely independent: since they are infinitely many values, the probability of one being beneath whatever bound I take will be 1.

One example which does work is when the $f_i$ are guaranteed to be k-lipschitz. Then, you can only consider the values of the sum on a sufficiently fine-grained grid, since there are finitely many values, you know that the sum grow linearly, and from the sum being nk-lipschitz, you can show that the sum grows everywhere linearly (though slower than $\mu$ in general) While this works, this seems like a too strong condition, and I'm hoping you can maybe help me find more relaxed conditions

• What are the events analogous to $\{S_n\geqslant n(\mu-\varepsilon)\}$ that you have in mind in the infinite-dimensional setting?