The paper by Longstaff-Schwatz on Least Square Monte Carlo offers very little proof. The only proof they have given assumed the option can only be exercised at two different time point and the price dynamics is supported on $(0,\infty)$ and is Markovian.
I have two questions
Are there further numerical studies on LSM for multiple variables?
Has people proven made any more progress proof-wise? (or do practitioners simply not care and cross their fingers?)