Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$.

Now let $X_1,\dots,X_t$ be a sequence of samples drawn independantly from $F$.

Let $$G_t(s) = \frac{1}{t}\sum_{i=1}^t \max(0,u(X_i)-sX_i)$$ and let $s_t$ be the unique fixed point of $G_t$.

Is it true that $s_t$ converges almost surely to $s_0$, and is it possible to say something about the speed of convergence? If not, how could I construct a converging estimator of $s_0$?

What I know is true is that due to the strong law of large number, for all $s$, $G_t(s)\to G(s)$ almost surely.

Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$.

Now let $X_1,\dots,X_t$ be a sequence of samples drawn independantly from $F$.

Let $$G_t(s) = \frac{1}{t}\sum_{i=1}^t \max(0,u(X_i)-sX_i)$$ and let $s_t$ be the unique fixed point of $G_t$.

Is it true that $s_t$ converges almost surely to $s_0$, and is it possible to say something about the speed of convergence? If not, how could I construct a converging estimator of $s_0$?