Uniform convergence of averages for stationary ergodic process Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process. 
I'm interested in the uniform convergence of averages: 
$$
\sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \mathbb{E}[X_0]\right|\to 0, n\to \infty,
$$
for some $R_n\gg n$. Are there any results of this type?

$^*$Precisely, I'm looking at the exponent of the so-called shot-noise potential:
$$
X_t = \exp\left\{\sum_{x\in \Pi} \phi(x-t)\right\},
$$
where $\Pi$ is a Poisson point process, and $\phi$ can be assumed as good as needed (e.g. continuous with finite support). 
 A: Uniform convergence holds when $R_n$ is at most a power of $n$. 
Using the tail of a Poisson variable, you can easily infer that $P(X_t>r) \le r^{-C\log \log r}$ for some $C$ that depends on the maximum and the finite support of $\phi$.
Thus when $\phi$ has finite support, $X_t$ has finite moments of all orders.
The $2k$'th moment of
$$
S_x:= \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \mathbb{E}[X_0]\right|
$$
can then be bounded by $C_k n^{-k}$, where $C_k$ depends on $k$ and $\phi$. Therefore 
$P(S_x>\epsilon) \le C_k (n\epsilon)^{-k}$.
From this, one can apply chaining (see Talagrand's book  springer.com/gp/book/9783642540745 )
to prove uniform convergence  in the original formulation,
provided $R_n/n^{k} \to 0$ as $n \to 0$.  
A: I found an elementary way to proceed through Chebyshev inequality. 
Assume that $\phi\in C(\mathbb{R})$ and $|\phi(x)|\le \frac{C}{1+|x|^{1+\beta}}$ for some $\beta>0$. It is known (Carmona-Molchanov 1995) that for any $\delta>0$, $$X_t = o(|t|^\delta), t\to \infty,\tag{1}$$ a.s. It is also not hard to see that $$
\operatorname{var}\left(\frac1{2n}\int_{-n}^{n} X_t dt\right) = O\Bigl(\frac1n\Bigr), n\to\infty.
$$
Now take some $r\in (1,2)$ and $a\in (r-1,1)$ and consider $A_n = \{k n^{r-a}, k=-[n^{a}],\dots,[n^a]+1\}$. Since $r-a<1$ and thanks to $(1)$, the average does not change a lot between the points of $A_n$, so for any $\varepsilon>0$,
$$
\limsup_{n\to\infty}\mathrm{P}\left(\sup_{|x|\le n^r} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \mathbb{E}[X_0]\right|>\varepsilon\right) \\
= \limsup_{n\to\infty}\mathrm{P}\left(\sup_{|x|\in A_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \mathbb{E}[X_0]\right|>\varepsilon\right)\\
\le \limsup_{n\to\infty} \sum_{x\in A_n} \mathrm{P}\left(\sup_{|x|\in A_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \mathbb{E}[X_0]\right|>\varepsilon\right)\\
\le \limsup_{n\to\infty}\frac{cn^a}{\varepsilon^2} \operatorname{var}\left(\frac1{2n}\int_{-n}^{n} X_t dt\right)=0. 
$$
(With a little bit more effort an almost sure convergence can be shown for any $r>1$, but the above is enough for me.)
