Following a previous question (here you'll find an introduction):
The book states that using the convergence of the binomial distribution towards the Poisson distribution, it's easy to show that $$|\{x\le\xi:\pi_S(x+\lambda \log x)-\pi_S(x)=k\}|\sim\xi\mathrm e^{-\lambda} \frac{\lambda^k}{k!}\quad(\xi\to\infty)$$ holds almost surely.
I couldn't prove this.