Is $E[g(x_i)g(x_j)]=E[g(x_i)]\,E[g(x_j)]$, for $x_i$ multinomial?

Question

Consider the multinomial distribution (Wikipedia):

and let $g:\mathbb{R}\longrightarrow \mathbb{R}^+$ a smooth function. I would like to know if one can show the identity

$$E[g(x_i)g(x_j)]=E[g(x_i)]\,E[g(x_j)],$$

for every $i\neq j$ fixed.

Answer 1

Let $k=2$, and $n=1$. Let $g$ the identity map.

We have $$X_1+X_2=1$$

then $$E[X_1X_2]=E[X_1(1-X_1)]=E[X_1]-E[X_1^2]=E[X_1]-E[X_1]=0$$

but $E[X_i] \ne 0$.