Simple random walk with absorbing boundary

[forza_bologna]

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.

 

[Hemant Rupani]

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.