[ergodicity]11_error_in_solution2 (More Discussion Required)

[shlff]

Dear @jstac , for solution 2 of lecture ergodicity, this PR corrects a mistake and adds a condition to clarify its holding:

• $(\phi Q)(j) = \sum_{i \geq 0} \phi(i) Q(i, j) = - \lambda_j \phi(j) + \lambda_{j} \phi(j) = 0$ -->> $(\phi Q)(j) = \sum_{i \geq 0} \phi(i) Q(i, j) = - \lambda_j \phi(j) + \lambda_{j-1} \phi(j-1) = 0$
• Then -->> Then, for any $j \geq 1$,

Note that $\phi Q$ is a forward equation and because of the definition of Q, we have the corrected math expression above.

However, this brings us a new issue that the statement after this math expression, It follows that $\phi$ is constant on $\ZZ_+$., may not hold.

• This statement holds if we consider a special case of the pure birth process: when $\lambda_i=\lambda$ for all $i$.

I am still thinking about a way to fix the new issue, and this may require further discussion and corrections.

@jstac , what do you think?

[jstac]

Good catch @shlff , well done!

I agree with your assessment. I will merge this, but could you please open an issue in the issue tracker explaining about this problem?

[shlff]

Thanks, @jstac . Sure!