In Chung's book, he defines the stopping time $ \alpha^k $ in the following way.
- $\alpha^1 = \alpha$;
- $\alpha^{k+1}(\omega) = \alpha^k(\tau^\alpha \omega)$;
where $\tau^\alpha$ is the $\alpha$-shift. Then he defines $ \beta_k = \sum_{j=1}^k \alpha^j$.
When I go through the inductive definition of $\alpha^k$, I get confuse. For example,
when $ k = 2 $, $\alpha^2(\omega) = \alpha^1(\tau^\alpha \omega)$;
when $ k = 3 $, $\alpha^3(\omega) = \alpha^2(\tau^\alpha \omega) = \alpha^1(\tau^\alpha(\tau^\alpha \omega)) = \alpha^1(\tau^{\alpha+\alpha} \omega)= \alpha^1(\tau^{2\alpha} \omega)$??
What should be the vaule of $\alpha$ of the shift $\tau$ in each case?
In else where I find the following definition:
$\alpha_k = \alpha_{k-1} + \alpha(\tau^{\alpha_{k-1}} \omega)$.
I think the above $\alpha_k$ is the same as $ \beta_k$ in Chung's book.
Can anyone help? Thanks very much in advance.