Details of proof of convergence of Bernstein polynomial approximation

Question

In Durrett's Probability Theory and Examples https://services.math.duke.edu/~rtd/PTE/PTE4_1.pdf, he gives the following.

I have two questions:

1. how do we get the underlined line?
2. We apply Jensen's inequality, but how do we know that we have convexity?

Answer 1

Recall that

$$\mathbb{E}g(X) = \sum_{m=0}^N \mathbb{P}(X=x_m) g(x_m)$$

for any discrete random variable $X$ taking values in $\{x_0,\ldots,x_N\}$. Applying this formula to $X := S_n$ and $g(x) := f(x/n)$ we find

$$\mathbb{E}f(S_n/n) = \sum_{m=0}^n \mathbb{P}(S_n=m) f(m/n) = \sum_{m=0}^n {n \choose m} p^m (1-p)^{n-m} f(m/n)=f_n(p).$$

Regarding your second question: Durrett is using Jensen's inequality for the convex function $g(x) := |x|$, i.e. he is using that

$$|\mathbb{E}(Y)| \leq \mathbb{E}(|Y|)$$

for

$$Y:= f \left( \frac{S_n}{n} \right)-f(p).$$

Answer 2

1. Using the law of the unconscious statistician : $$Ef(S_n/n)=\sum_{n\geq m\geq0} f(m/n) P(S_n=m)=\sum_{n\geq m\geq0} {n \choose m}p^m(1-p)^mf(m/n)=f_n(p).$$
2. Let $\phi(x)=|x-f(p)|$, then $\phi$ is convex and, by Jensen's inequality : $$|Ef(S_n/n)-f(p)|=\phi(Ef(S_n/n)) \leq E\left[\phi(f(S_n/n))\right]=E|f(S_n/n)-f(p)|.$$