Let $f_n(t)$ be a sequence of random convex functions on $\mathbb R$. Assume they're differentiable, have uniformly bounded derivatives, and have finite second moments. Denote by $g_n(t):= \mathbb E f_n(t)$ the expectation (which is also convex and differentiable). Suppose
there exists $\lim_{n\to\infty} g_n(t) =: g(t)\in\mathbb R$
there exists $\lim_{n\to\infty} \mathbb E \big(f_n(t) - g_n(t)^2)^2 = 0$
for every $t\in\mathbb R$. I have to prove the following:
- there exists $\lim_{n\to\infty} g_n'(t) = g'(t)\in\mathbb R$
- there exists $\lim_{n\to\infty} \mathbb E \big( f_n'(t) - g_n'(t)^2 \big)^2 = 0$
for almost every $t\in\mathbb R$, precisely for those $t$ such that $g'(t)$ exists (which are all but a countable number).
The answer to 1. should be already contained in Limit of derivatives of convex functions and A convex function is differentiable at all but countably many points .
For the proof of 2. I think I have to use convexity, maybe the fact that the incremental ratio is increasing, but I don't know how to procede. Can you help me?
