In these notes: https://johnthickstun.com/docs/bscrr.pdf, towards the end of the proof of Proposition 5.2 on page 6, the author claims:

$$ \log \lim_{n \to \infty} \Bbb{E}_\pi \left[\frac {S^*_n} S \right] = \log \Bbb{E}_\nu \left[\frac {S^*} S \right] $$

I.e., that

$$ \lim_{n \to \infty} \mathbb{E}_\pi \left[\frac {S^*_n} S \right] = \mathbb{E}_\nu \left[\frac {S^*} S \right] $$

I'm not entirely following why that's true. In Proposition 4.2, the author does prove that

$$ \log \frac {S^*_n} S \xrightarrow [n \to \infty] {d} \mathcal{N}(\nu T, \sigma^2 T) $$

in distribution (weakly), but that's weaker than convergence in mean, which is what he seems to be using. He does refer to Proposition 4.1, but Proposition 4.1 talks about the log-returns, not the returns, and in any case, doesn't speak to the limit. So even if I try to compute the expected value of $\frac {S^*_n} S$ with respect to the risk-neutral probability measure $\pi$ directly, it ends up being $\mathbb{E}_\pi \left[\frac {S^*_n} S \right] = T \log r$, which he already has. What am I missing?



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