Probability
The Analysis of Data, volume 1
Important Random Processes: Exercises
$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$
7.5. Exercises
- Prove formally that all first and second-dimensional marginals of a Markov chain are specified by the transition probabilities $T$ and the initial probabilities $\rho$.
- Prove formally that the following equation follows from the Markov property of a Markov chain
\begin{multline*}
\P(X_1=x_1,\ldots,X_n=x_n,X_{n+1}=x_{n+1},\ldots,X_{n+m}=x_{n+m}| X_0=x_0) \\
= \P(X_1=x_2,\ldots,X_n=x_n|X_0=x_0) \P(X_{1}=x_{n+1},\ldots,X_{m}=x_{n+m}| X_0=x_n).
\end{multline*}
- Show that a non-symmetric random walk on $\mathbb{Z}$ (Example 7.1.4) with $d=1$ and probability of moving forward being different from the probability of moving backward) is transient.