Question: Is there a $p$-adic analogue of the Strong Law of Large Numbers? In particular, suppose that $f\_i: \mathbb{Z}\_p \longrightarrow \mathbb{Q}\_p$ for $i = 1,2,\ldots$ is an sequence of random variables that is i.i.d. with respect to the $\mathbb{Q}\_p$-valued normalized Haar measure $\mu$ on $\mathbb{Z}\_p$. Must one have $$\int\_{\mathbb{Z}\_p} f\_1 d \mu = \lim\_{n \rightarrow \infty} \frac{f\_1(x) + f\_2(x) + \cdots + f\_n(x)}{n}$$ for almost all $x \in \mathbb{Z}\_p$? If not, is there a partial result in this direction?

Answer: 
I think this can't work. If you look at the simplest possible case, where the $f\_i$ take values 0 and 1 with probabilities $1-1/p$ and $1/p$, say, then you are effectively looking at the average of the digit sums of $x$. 

To make sense (since your $f\_i(x)$ generally take values in $\mathbb Q\_p$), you should look for a limit in $\mathbb Q\_p$. But now, the numerator is an integer close to $n/p$. By standard random walk techniques, you know that the ratio will be exactly $1/p$ infinitely often (with $p$-valuation $-1$).

On the other hand, for $n$ not a multiple of $p$, the $p$-valuation of the ratio will be (at least) $0$. Hence there is no pointwise convergence.
There seems to be an extensive oeuvre of one A. Khrennikov on the subject. Here is [one paper.](https://www.dropbox.com/s/i43crfb873xs78s/khren2.pdf?dl=0) It seems one can get interesting results (his interest comes from mathematical physics).