Probability
The Analysis of Data, volume 1
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}}{=}}
$
6.8. Exercises
- Describe a collection of finite dimensional marginals that violates the consistency conditions.
- Prove formally that the marginals defining the iid process satisfy the consistency conditions.
- Prove that the random walk process $\mathcal{Z}$ has one dimensional binomial marginals.
- Consider a discrete-time random walk model similar to the on in this chapter, with the following exception: the walker steps two steps forward with probability $\theta$ and one step backwards with probability $1-\theta$. Characterize the process in terms of the RP categories (stationary, Markov, etc.), and derive its expectation, variance, and autocorrelation functions.