Independence of random variable and sum of iid random variables

Question

Let $T_n=\sum_{i=1}^{n} X_i$ and $\{ X_i \} $ be a sequence of i.i.d. (strictly) positive random variables.

So I know that $X_{n+1}$ is independent of $X_1,...,X_n$. Futher we have $T_{n+1}=T_n+X_{n+1}$.

Is it then true, that $X_{n+1}$ is also independent of $T_n=X_1+...+X_n$?

Answer 1

Yes. If $X$ and $Y$ are independent random variables, and $f, g$ are (measureable) functions, then we have that $f(X)$ and $g(Y)$ also are independent random variables. This generalizes to functions of multiple variables, which gives your case.