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Chapter 6: Problem 1

A random walk on the set \(\\{0,1,2, \ldots, b\\}\) has transition matrix given by \(p_{00}=1-\lambda_{0}, p_{b b}=\) \(1-\mu_{b}, p_{i, i+1}=\lambda_{i}\) and \(p_{i+1, i}=\mu_{i+1}\) for \(0 \leq i

Short Answer

Expert verified
The process satisfies the detailed balance condition, confirming reversibility in equilibrium.

Step by step solution

01

Understanding Reversibility

A stochastic process is reversible if the detailed balance condition holds, which means \( \pi_i p_{i,j} = \pi_j p_{j,i} \) for all states \(i\) and \(j\). This condition ensures that in equilibrium, the flow from state \(i\) to state \(j\) is equal to the flow from state \(j\) to state \(i\).
02

Setting Up the Transition Probabilities

For the given random walk, we have the transition probabilities: \( p_{i, i+1} = \lambda_i \) and \( p_{i+1, i} = \mu_{i+1} \). We also know that \( \lambda_i + \mu_i = 1 \). This information is critical for establishing the balance condition.
03

Define the Equilibrium Distribution

Assume an equilibrium distribution \( \pi = (\pi_0, \pi_1, \ldots, \pi_b) \). At equilibrium, the probabilities must satisfy \( \pi_i p_{i,i+1} = \pi_{i+1} p_{i+1,i} \). Begin by setting up these equations for a generic state \(i\).
04

Solving for Equilibrium Ratios

Use the detailed balance condition: \( \pi_i \lambda_i = \pi_{i+1} \mu_{i+1} \). Solving this gives the ratio \( \frac{\pi_{i+1}}{\pi_i} = \frac{\lambda_i}{\mu_{i+1}} \), a key expression for solving the equilibrium probabilities for all states.
05

Inductive Completion of Equilibrium Probabilities

Express \( \pi_i \) in terms of \( \pi_0 \) using the ratios found: \( \pi_i = \pi_0 \prod_{k=0}^{i-1} \frac{\lambda_k}{\mu_{k+1}} \). This step utilizes induction to show how each probability \(\pi_i\) depends on \(\pi_0\) and the transition probabilities.
06

Normalizing the Probabilities

Ensure that the total probability sums to 1. Calculate \( \pi_0 \) by solving \( 1 = \sum_{i=0}^{b} \pi_i = \pi_0 \left( 1 + \sum_{i=1}^{b} \prod_{k=0}^{i-1} \frac{\lambda_k}{\mu_{k+1}} \right) \). This constraint will identify the value for \( \pi_0 \) and hence all other \( \pi_i \)'s.
07

Conclusion on Reversibility

By satisfying the detailed balance conditions and ensuring that \(\pi_i\) are valid probabilities, the process is shown to be reversible in equilibrium. The collapses of the equations \( \pi_i p_{i,i+1} = \pi_{i+1} p_{i+1,i} \) confirm reversibility for each segment in the state space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reversible Process
In the world of random walks, a reversible process holds a special place. Think of it as a balanced dance, where every step forward is matched by a step backward. But what exactly does this mean in mathematical terms? A random walk is considered reversible if, for every pair of states, the process can go between them equally in both directions. This is formalized by the
  • Detailed Balance Condition: \( \pi_i p_{i,j} = \pi_j p_{j,i} \)
This condition tells us that in equilibrium, the flow of probability from state \(i\) to state \(j\) should be the same as the flow from \(j\) back to \(i\). It's like saying that every transition to the left is matched by an equal likelihood transition to the right, keeping the system balanced. In our random walk problem, confirming that this condition is met ensures the process is indeed reversible.
Equilibrium Distribution
The term 'equilibrium distribution' might sound a bit intimidating, but let's break it down. In simple terms, an equilibrium distribution for a random walk is a stable probability distribution across all possible states such that the state probabilities do not change over time. At this point:
  • Each state's probability remains constant.
  • The overall system is in balance.
Finding the equilibrium distribution is crucial because it gives us a snapshot of the system when it has reached a state of steadiness.
In our exercise, the equilibrium distribution \( \pi = (\pi_0, \pi_1, \ldots, \pi_b) \) must meet the condition:
  • \( \pi_i p_{i,i+1} = \pi_{i+1} p_{i+1,i} \)
This equation is a cornerstone of ensuring our random walk balances out over time. Solving these equations helps in calculating each state's probability, providing a deeper understanding of how the system behaves at equilibrium.
Detailed Balance Condition
The detailed balance condition is a gem of a concept in the study of random processes. It is the heart of reversibility. This condition is not just a rule; it's the very glue that holds reversible processes together. In essence, it provides a simple yet pivotal equation to check:
  • \( \pi_i p_{i,j} = \pi_j p_{j,i} \)
The detailed balance condition ensures that, at equilibrium, transitions between any two states are in perfect harmony. It's a neat symmetry where the probability from state \(i\) to state \(j\) equals the probability from \(j\) back to \(i\), weighted by their respective equilibrium probabilities \( \pi_i \) and \( \pi_j \). In our previous steps, setting up these balance equations and solving for state probabilities is how we confirmed the reversibility of the random walk.

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Most popular questions from this chapter

(a) Show that for cach pair \(i, j\) of states of an irreducible aperiodic chain, there exists \(N=N(i, j)\) such that \(p_{i j}(n)>0\) for all \(n \geq N\). (b) Let \(X\) and \(Y\) be independent irreducible aperiodic chains with the same state space \(S\) and transition matrix P. Show that the bivariate chain \(Z_{n}=\left(X_{n}, Y_{n}\right), n \geq 0\), is irreducible and aperiodic. (c) Show that the bivariate chain \(Z\) may be reducible if \(X\) and \(Y\) are periodic.

Consider the symmetric random walk in three dimensions on the set of points \((x, y, z): x, y, z=\) \(0, \pm 1, \pm 2, \ldots, 1 ;\) this process is a sequence \(\left[\mathbf{X}_{n}: n \geq 0\right]\) of points such that \(\mathbb{P}\left(\mathbf{X}_{n+1}=\mathbf{X}_{n}+\epsilon\right)=\frac{1}{6}\) for \(\epsilon=(\pm 1,0,0),(0, \pm 1,0),(0,0, \pm 1)\). Suppose that \(\mathbf{X}_{0}=(0,0,0)\). Show that $$ \mathrm{P}\left(\mathbf{X}_{2 n}=(0,0,0)\right)=\left(\frac{1}{6}\right)^{2 n} \sum_{i+j+k=n} \frac{(2 n) !}{(i) j ! k !)^{2}}=\left(\frac{1}{2}\right)^{2 n}\left(\begin{array}{c} 2 n \\ n \end{array}\right) \sum_{i+j+h=n}\left(\frac{n !}{3^{n} i 1 j ! k !}\right)^{2} $$ and deduce by Stirling's formula that the origin is a transient state.

You stick pins in a Mercator projection of the Farth in the manner of a Poisson process with constant intensity \(\lambda\). What is the intensity function of the corresponding process on the globe? What would be the intensity function on the map if you formed a Poisson process of constant intensity \(\lambda\) of meteorite strikes on the surface of the Earth?

Let \(\left(X_{n}: n \geq 0\right)\) be a persistent irreducible discrete-time Markov chain on the state space \(S\) with transition matrix \(\mathbf{P}\), and let \(x\) be a positive solution of the equation \(x=\mathbf{x P}\) (a) Show that $$ q_{i j}(n)=\frac{x_{j}}{x_{i}} p_{j i}(n), \quad i, j \in S, n \geq 1 $$ defines the \(n\)-step transition probabilities of a persistent irreducible Markov chain on \(S\) whose first-passage probabilities are given by $$ g_{i j}(n)=\frac{x_{j}}{x_{i}} l_{f i}(n), \quad i \neq j, n \geq 1 $$ where \(l_{j i}(n)=\mathrm{P}\left(X_{n}=i, T>n \mid X_{0}=j\right)\) and \(T=\min \left(m>0: X_{m}=j\right\\}\) (b) Show that \(\mathrm{x}\) is unique up to a multiplicative constant. (c) Let \(T_{j}=\min \left(n \geq 1: X_{n}=j\right)\) and define \(h_{i j}=\mathrm{P}\left(T_{j} \leq T_{i} \mid X_{0}=i\right) .\) Show that \(x_{i} h_{i j}=x_{j} h_{j i}\) for all \(i_{+} j \in S\).

Let \(\lambda \mu>0\) and let \(X\) be a Markov chain on \((1,2)\) with generator $$ \mathbf{G}=\left(\begin{array}{cc} -\mu & \mu \\ \lambda & -\lambda \end{array}\right) $$ (a) Write down the forward equations and solve them for the transition probabilities \(p_{i j}(t), i, j=\) I, 2. (b) Calculate \(\mathbf{G}^{n}\) and hence find \(\sum_{n=0}^{\infty}\left(t^{n} / n !\right) \mathbf{G}^{n}\). Compare your answer with that to part (a). (c) Solve the equation \(\pi \mathrm{G}=0\) in order to find the stationary distribution. Verify that \(P_{i j}(t) \rightarrow \pi_{j}\) as \(t \rightarrow \infty\).
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