Let $\left(X_k\right)_{k\geqslant 1}$ be a independent sequence of positive identically distributed random variables, $Y_k = \frac{X_{2k-1}}{X_{2k-1}+X_{2k}}$, and $S_n = Y_{1} + \cdots + Y_{n}$. Is it true that $\frac{S_{n}}{n} \to a$ a.s. for some constant $a$?
I'm confident that I know the strategy; first prove $Y_k$ are independent, identically distributed and that $\mathbb{E}[Y_1] < \infty$. Then invoke Kolmogorov's Strong Law of Large numbers which tell us $\frac{S_n}{n} \to \mathbb{E}[Y_1]$. I'm having trouble rigorously proving the following:
How can we prove $Y_k$ are independent and identically distributed?
How can we actually find $\mathbb{E}[Y_1]$?
For the first question my idea is that if we take $Y_1$ and $Y_2$ for example, then since $X_1, X_2$ are independent from $X_3, X_4$ then it follows $\frac{X_3}{X_3 + X_4}$ is independent from $\frac{X_1}{X_1 + X_2}$. But is this necessarily true? How could I rigorously prove it?
Thank you very much.