Consider a Markov chain with state space {0,1,2,3} and a...

Consider a Markov chain with state space $\{0,1,2,3\}$ and a transition matrix so $P_{0,3}=3 / 5$ is the probability of moving from state 0 to state $=\left[\begin{array}{cccc}0 & 3 / 10 & 1 / 10 & 3 / 5 \\ 1 / 10 & 1 / 10 & 7 / 10 & 1 / 10 \\ 1 / 10 & 7 / 10 & 1 / 10 & 1 / 10 \\ 9 / 10 & 1 / 10 & 0 & 0\end{array}\right]$ ility of moving from state 0 to state 3 . so $P_{0,3}=3 / 5$ is the probability of moving from state 0 to (b) Find the probability of being in state 3 after 32 steps if the chain begins at state 0 . (c) Find the probability of being in state 3 after 128 steps if the chain begins at a state chosen uniformly at random from the four states. (d) Suppose that the chain begins in state 0 . What is the smallest value of $t$ for which $\max _{s}\left|P_{0, s}^{t}-\pi_{*}\right| \leq 0.01 ?$ Here $\tilde{\pi}$ is the stationary distribution. What is the smallest value of $t$ for which $\max _{3}\left|P_{0}^{2}-\pi_{2}\right|<0,001$ ?

Consider a Markov chain with state space $\{0,1,2,3\}$ and a transition matrix
so $P_{0,3}=3 / 5$ is the probability of moving from state 0 to state
$=\left[\begin{array}{cccc}0 & 3 / 10 & 1 / 10 & 3 / 5 \\ 1 / 10 & 1 / 10 & 7 / 10 & 1 / 10 \\ 1 / 10 & 7 / 10 & 1 / 10 & 1 / 10 \\ 9 / 10 & 1 / 10 & 0 & 0\end{array}\right]$ ility of moving from state 0 to state 3 .
so $P_{0,3}=3 / 5$ is the probability of moving from state 0 to
(b) Find the probability of being in state 3 after 32 steps if the chain begins at state 0 .
(c) Find the probability of being in state 3 after 128 steps if the chain begins at a state chosen uniformly at random from the four states.
(d) Suppose that the chain begins in state 0 . What is the smallest value of $t$ for which $\max _{s}\left|P_{0, s}^{t}-\pi_{*}\right| \leq 0.01 ?$ Here $\tilde{\pi}$ is the stationary distribution. What is the smallest value of $t$ for which $\max _{3}\left|P_{0}^{2}-\pi_{2}\right|<0,001$ ?

Key Concepts

Total Variation Distance

Total variation distance is a measure of the discrepancy between two probability distributions. In the context of Markov chains, it is used to quantify how far the chain’s current state distribution is from its stationary distribution, assisting in determining the mixing time.

Mixing Time

Mixing time refers to the number of steps required for the Markov chain to become close to its stationary distribution, regardless of the starting state. It is typically measured by how small the difference (commonly in total variation distance) between the current distribution and the stationary distribution becomes over time.

Stationary Distribution

A stationary distribution is a probability distribution over the state space that remains unchanged by the transition matrix. In other words, if the chain starts in the stationary distribution, it will remain in that same distribution after any number of steps, and it often represents the long-term behavior of the Markov chain.

Transition Matrix

The transition matrix of a Markov chain is a square matrix where each entry represents the probability of transitioning from one state to another. This matrix encapsulates all the dynamics of the Markov process, with each row summing to 1, reflecting that the total probability of moving from a given state to any state is 100%.

Markov Chain

A Markov chain is a stochastic process in which the future state of the system depends only on its present state, not on the sequence of events that preceded it. This memoryless property simplifies analysis since the process can be fully described by its current state and transition probabilities.

n-step Transition Probabilities

n-step transition probabilities are obtained by raising the transition matrix to the nth power. This operation yields the probabilities of moving from one state to any other state in exactly n steps, allowing analysis of the chain's behavior over time.

Please give Ace some feedback