4 votes
Accepted

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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  • 10.6k
3 votes

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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  • 2,775
2 votes

Probability of a two-state continuous Markov chain

Yes, the calculation in the discrete time case is explained on page 3 of [1], and it works similarly in the continuous time case. If $u_t=P(\epsilon_t=0 | \epsilon_0=0)$ then $$u_t'=p_0(1-u_t)-p_1u_t \...
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1 vote
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Two definitions of a Stochastic Process?

In the definition you start with, $$\tag{1} (y(x_1),...,y(x_n)) \sim \mathcal{N}(\mathbf{\mu}_n,\mathbf{\Sigma}_n),\quad (\forall \; n \in \mathbb{N} \text{ and } x_1,...,x_n). $$ we need labels at ...
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  • 4,254
1 vote
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Distribution of solution to SDE

From the general solution to linear SDEs, you have : $$X_t=e^{-\frac{1}{2}t}\left(X_0+\int_0^t e^{\frac{1}{2}s}dB_t\right)$$ So what do we have here ? A deterministic term $e^{-\frac{1}{2}t}X_0$ and ...
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  • 5,382
1 vote
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Fubini's theorem and time integrals of stochastic processes

Your first application of Fubini's theorem is correct, but the second one does not seem to hold in general. First application Recall that to apply Fubini's theorem to $f$, you must have $f \in L^1( \...
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  • 10.2k
1 vote

Write probability of first return time in terms of first hitting time

I'd write like this: \begin{align} P_i(T_i < \infty) &= \sum_{j \in I}P_i(T_i < \infty \mid X_1 = j)P_i(X_1 = j) \end{align} Now, in order to use the Markov property, we write $T_i$ as $T_i(...
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  • 6,857
1 vote
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Time series: ARMA characteristic polynomials have common roots

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-\...
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  • 10.2k
1 vote
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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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