Question

It follows from the argument made in Exercise 38 that state \(i\) is null recurrent if it is recurrent and \(\pi_{i}=0 .\) Consider the one-dimensional symmetric random walk of Example 4.18. (a) Argue that \(\pi_{i}=\pi_{0}\) for all \(i\). (b) Argue that all states are null recurrent.

Short answer

In a one-dimensional symmetric random walk, the stationary distribution \(\pi_i\) is constant for all states \(i\). So, \(\pi_i = \pi_0\) for all states \(i\). Since all states are recurrent and infinite in number, the normalization condition (sum of \(\pi_i\) over all states should be equal to 1) can only be satisfied if \(\pi_0 = 0\). Thus, all states are null recurrent, as they are recurrent with a stationary distribution of \(\pi_i = 0\).

Step-by-step solution

Understanding One-Dimensional Symmetric Random Walk

In a one-dimensional symmetric random walk, a particle moves either to the left or to the right with equal probability. At each step, the particle has a probability of 0.5 to move in either direction.

Part (a): Argue that \(\pi_i = \pi_0\) for all i

A state \(i\) is said to be recurrent if the particle is expected to return to this state infinite number of times. We know that state \(i\) is recurrent if the sum of the probabilities to the left and right of the state is equal. We also know that, in a symmetric random walk, the particle moves either to the left or to the right with equal probability, which is 0.5. So for all states \(i\), we have \[\pi_i = \frac{1}{2} \pi_{i-1} + \frac{1}{2} \pi_{i+1}\] This equation shows the balance condition for the symmetric random walk. Since we have equal probabilities for moving left and right, we can argue that the stationary distribution \(\pi_i\) is constant for all states \(i\). Thus, we have \[\pi_i = \pi_0\] for all states \(i\).

Part (b): Argue that all states are null recurrent

To argue that all states are null recurrent, we need to show that they are recurrent and that the stationary distribution \(\pi_i = 0\) for all i. From part (a), we already know that all states have the same stationary distribution \(\pi_i = \pi_0\). In a one-dimensional symmetric random walk, we also know that the particle is expected to return to each state infinite number of times, meaning that every state is recurrent. In order to satisfy the normalization condition (sum of \(\pi_i\) over all states should be equal to 1), we need to find a value for \(\pi_0\). However, since there are an infinite number of states in a one-dimensional random walk, the sum of the stationary distribution \(\pi_i\) over all states would be infinite if \(\pi_0 > 0\). This would violate the normalization condition. Therefore, we must have \[\pi_0 = 0\] Thus, all states are recurrent, and their stationary distribution is \(\pi_i = 0\), meaning that all states in a one-dimensional symmetric random walk are null recurrent.