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Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, non-negative constants $\epsilon_k$. Show that:

(1) If $\lim_{n\to\infty} \epsilon_n=0$ then $S_n/n\to m$ in $L^2$ and probability

(2) If $\sum_{k=1}^\infty \epsilon_k<\infty$, then $\mathrm{Var}(S_n/n)$ is of order $O(1/n)$ and $S_n/n\to m$ almost surely

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(1) First of, I found a similar looking question here, but we don’t have, that the constants are bounded by $1$, so I don’t know if the Chebyshev-inequality approach works here.

(2) We have $\mathrm{Var}(S_n/n)=\dfrac{1}{n^2}Var(S_n)=\dfrac{1}{n^2}\sum_{i\ne j}\mathrm{Cov}(X_i,X_j)=\dfrac{1}{n^2}(\sum_{k=1}^n\mathrm{Cov}(X_i,X_i)+\sum_{i=1}^{n-1}\sum_{j=i}^n \mathrm{Cov}(X_i,X_j))\leq\dfrac{1}{n^2}(n\cdot\epsilon_0+\sum_{i=1}^{n-1}\sum_{j=i}^n \mathrm{Cov}(X_i,X_j))$.

This is where I’m stuck.

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We can assume that $m=0$. Then take $\delta\gt 0$ write for a fixed $C$:

$$\frac 1{n^2} \mathbb E[S_n^2]\leqslant \frac 1{n^2}\sum_{i,j=1}^n\varepsilon_{|i-j|}

[|i-j|\leqslant C]+\frac 1{n^2}\sum_{i,j=1}^n\varepsilon_{|i-j|}

[|i-j|\gt C].$$

If we choose it such that $\varepsilon_k \leqslant \delta$ if $k\geqslant C$, then we have

$$\frac 1{n^2} \mathbb E[S_n^2]\leqslant \frac{2Cn}{n^2}\sup_k \varepsilon_k+\delta,$$

hence the sequence $\left(\mathbb E[S_n^2]/n^2\right)_{n\geqslant 1}$ converges to $0$.

Once we showed the good asymptotic of $\mathbb E[S_n^2]$, an application of the Borel-Cantelli’s lemma yields $2^{-n}S_{2^n}\to 0$ almost surely. Indeed, define for a fixed $j$ the event $A_n:=\{|S_{2^n}|\gt 2^n/j\}$ for a fixed $\delta$. Then the series $\sum_n\mathbb P(A_n)$ is convergent, and we get $\mathbb P\left(\limsup_{n\to\infty} A_n \right)=0$. As a consequence, on a set $\Omega_j$ of probability one, we have $|S_{2^n}(\omega)| 2^n\lt1/j $ for $n\geqslant N(j,\omega)$.

In order to deduce the result for the whole sequence, note that

$$\mu\left\{\max_{2^n\leqslant k\lt 2^{n+1}}\frac{|S_k|}{2^n}>\delta\right\}

\leqslant 2^n\max_{2^n\leqslant k\lt 2^{n+1}}\mu\left\{\frac{|S_k|}{2^n}>\delta\right\}\\

\leqslant 2^n\max_{2^n\leqslant k\lt 2^{n+1}}\frac 1{\delta^2}\mathbb E\left[\frac{S_k^2}{2^{2n}}\right].$$

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