## Probability

### The Analysis of Data, volume 1

Important Random Processes: Exercises

## 7.5. Exercises

1. Prove formally that all first and second-dimensional marginals of a Markov chain are specified by the transition probabilities $T$ and the initial probabilities $\rho$.
2. 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*}
3. 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.