Let $X_1,X_2, \dots$ be independent Bernoulli random variables, $X_i \sim BIN(1,p_i)$ and let $$Y_n=\sum\limits_{i=1}^n (X_i-p_i)/n.$$ Show, using Chebychev inequality, that the sequence $Y_1, Y_2, \dots$ converges stochastically to $c=0$ as $n$ approaches infinity.
I am trying to use this as a step in another proof and I know it approaches $0$; just having trouble showing it using the Chebychev inequality. Any help would be appreciated!
We have that $$ \Pr\{|Y_n|>\varepsilon\}\le\frac{\operatorname EY_n^2}{\varepsilon^2} $$ for all $\varepsilon>0$. Using the independence and the fact that $0\le p_i\le 1$, \begin{align*} \operatorname EY_n^2 &=\frac1{n^2}\sum_{i=1}^n\operatorname E[X_i-p_i]^2\\ &=\frac1{n^2}\sum_{i=1}^n\operatorname E[X_i^2-2X_ip_i+p_i^2]\\ &=\frac1{n^2}\sum_{i=1}^n(\operatorname EX_i^2-2p_i\operatorname EX_i+p_i^2)\\ &=\frac1{n^2}\sum_{i=1}^n(p_i(1-p_i)+p_i^2-2p_i^2+p_i^2)\\ &=\frac1{n^2}\sum_{i=1}^np_i(1-p_i)\\ &=\frac1{n^2}\sum_{i=1}^n(p_i-p_i^2)\\ &\le\frac1{n^2}\sum_{i=1}^n(1-0)\\ &=\frac1n. \end{align*} Hence, $$ \Pr\{|Y_n|>\varepsilon\}\le\frac{1}{\varepsilon^2n}\to0 $$ as $n\to\infty$ for all $\varepsilon>0$.