Consider the following sequence of random variable: $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$, ... i.i.d with $E(X_{i}=\mu)$ and n even. Define: $P_{n}=\frac{2}{n}\sum\limits_{i=1}X_{2i}$ and $I_{n}=\frac{2}{n}\sum\limits_{i=1}X_{2i-1}$ the mean of the even and odd terms on the sample.
Suppose we know that: $\sqrt{n} (\overline{X_{n}}-\mu) \sim F$.
Show that $\sqrt{n} (P_{n}-\mu) \sim F$ and $\sqrt{n} (I_{n}-\mu) \sim F$
Any help?