Hello, I have a question related to chapter 3. There is a simple random walk on {0, 1, 2, ..., b} The boundary conditions are: P[\(X_{n+1}\) = 0 / \(X_{n}\) = 0] = \(\alpha\) P[\(X_{n+1}\) = 1 / \(X_{n}\) = 0] = 1 - \(\alpha\) The core reading states that all states are aperiodic unless \(\alpha\) is 1. If \(\alpha\) is 1, then the 0 boundary is absorbing. Then this is aperiodic or periodic? I do not understand how it can be periodic. Thank you.
Then the state 0 will not be periodic. as the core reading says, "all states are aperiodic unless both \(\alpha\) and \(\beta \) are either 0 or 1, ie unless \(\alpha\) = 0 or 1 and \(\beta \) = 0 or 1." ---That means not all states are aperiodic when both \(\alpha\) and \(\beta \) are either 0 or 1. there may be few aperiodic states.
