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?




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$^\*$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). 



Uniform convergence of averages for stationary ergodic process
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.)


