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### Can we control the distance between the empirical and theoretical mean on the whole trajectory any better than using Hoeffding and a union bound?

The dependence on $T$ is logarithmic, see $(4)$ below. I could not find a precise reference, so I give an argument below, making sure to get explicit constants. The argument is adapted from the ...
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### A statement about finite Markov chains

Let $N$ denote the number of visits to $y$ before returning to $z$. Let $I_t$ be the random variable that is 1 if $X_{t}=y, \tau_{z}^{+}>t$ and zero otherwise. Then by definition, $N = \sum_t I_t$. ...
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When writing down an ARMA process you should always eliminate common roots, as these roots are not identifiable (and thus not estimable). In your case you should indeed cancel out the common root $(1-\... • 10.2k 1 vote Accepted ### Question on interchanging of random variables with the same distribution inside expectation The substitution you made is wrong. Here is a general useful statement which helps in many situations including yours. Let$\mathcal{S}$be any sub-$\sigma\$-field. When you have an integrable random ...
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