Find the stationary distribution of a Markov chain X0, X1, X2, …on the state space {0,1, …, 110}

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Find the stationary distribution of a Markov chain X0, X1,...

Find the stationary distribution of a Markov chain $X_{0}, X_{1}, X_{2}, \ldots$ on the state space $\{0,1, \ldots, 110\}$, with transition probabilities given by $$ \begin{aligned} &P\left(X_{n+1}=j \mid X_{n}=0\right)=p, \text { for } j=1,2, \ldots, 110 ; \\ &P\left(X_{n+1}=0 \mid X_{n}=0\right)=1-110 p ; \\ &P\left(X_{n+1}=j \mid X_{n}=j\right)=1-r, \text { for } j=1,2, \ldots, 110 ; \\ &P\left(X_{n+1}=0 \mid X_{n}=j\right)=r, \text { for } j=1,2, \ldots, 110 \end{aligned} $$ where $p$ and $r$ are constants with $0<p<\frac{1}{110}$ and $0<r<1$.

Find the stationary distribution of a Markov chain $X_{0}, X_{1}, X_{2}, \ldots$ on the state space $\{0,1, \ldots, 110\}$, with transition probabilities given by
$$
\begin{aligned}
&P\left(X_{n+1}=j \mid X_{n}=0\right)=p, \text { for } j=1,2, \ldots, 110 ; \\
&P\left(X_{n+1}=0 \mid X_{n}=0\right)=1-110 p ; \\
&P\left(X_{n+1}=j \mid X_{n}=j\right)=1-r, \text { for } j=1,2, \ldots, 110 ; \\
&P\left(X_{n+1}=0 \mid X_{n}=j\right)=r, \text { for } j=1,2, \ldots, 110
\end{aligned}
$$
where $p$ and $r$ are constants with $0<p<\frac{1}{110}$ and $0<r<1$.

Key Concepts

Markov Chain

A Markov chain is a mathematical model describing a sequence of events in which the probability of each event depends only on the state attained in the previous event. This property, known as the Markov property, makes them particularly useful for modeling a wide array of real-world processes in a simplified manner.

Stationary Distribution

A stationary distribution of a Markov chain is a probability distribution over the states that remains unchanged as the chain evolves over time. It satisfies the property that, if the chain starts in the stationary distribution, it will have the same distribution at all future time steps. This is found by solving a system of balance equations derived from the transition probabilities of the chain.

Balance Equations

Balance equations, also known as equilibrium equations, are used to express the condition that the total probability entering each state is equal to the total probability leaving that state in the steady state. Solving these makes it possible to determine the stationary distribution for a Markov chain.

Transition Probability Matrix

The transition probability matrix is a matrix that records the probabilities of moving from one state to another in a Markov chain. Each entry in the matrix represents the probability of transitioning to a particular state from the current state, and it plays a crucial role in setting up the balance equations needed to find the stationary distribution.

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