Prove discrete Markov chain $x_i = \sqrt{1-\beta_i} x_{i-1} + \sqrt{\beta_i} z_{i-1}$ converges to $dx=-\frac{1}{2}\beta(t)xdt + \sqrt{\beta(t)} dw$.

Question

The following is the derivation taken from page 14 of the paper Score-Based Generative Modeling through Stochastic Differential Equations (https://arxiv.org/abs/2011.13456).

I am having a hard time following most of the proof. How to understand this derivation. Why rewriting the Markov Chain as Eq. (23) leads to the Eq. (24)?

This question should help people to understand better the theory behind Diffusion Process for Generative Modelling.

Erm can someone explain why this question cannot be published so that I can improve on it?