Alec's sample mean bound

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Needs some work, like what is a random variable for which expectation and variance are defined? Can we have complex or vector ones for example?

Inequality

Let $X_1,\ldots,X_n$ be a collection of $n$ random variables which are pairwise independent, such that:

• $\exists\mu\forall i\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mu\big]$ - all of the $X_i$ have the same expectation and
  • Alternatively: $\forall i,j\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mathbb{E}[X_j]\big]$, but note we need $\mu$ in the expression
• $\exists\sigma\forall i\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\sigma^2\big]$ - all the $X_i$ have the same variance
  • Alternatively: $\forall i,j\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\text{Var}(X_j)\big]$, but note again we need $\sigma$ in the expression

Then

• For all $\epsilon>0$ we have:
  • $$\mathbb{P}\left[\left\vert\frac{\sum^n_{i\eq 1}X_i}{n}-\mu\right\vert<\epsilon\right]\ge 1-\frac{\sigma^2}{\epsilon^2n}$$
    • Note that the notation here differs from that in my 2011 research journal slightly, but $\sigma$ and $\mu$ were present.
  • $$\mathbb{P}\left[\left\vert\frac{\sum^n_{i\eq 1}X_i}{n}-\mu\right\vert<\epsilon\right]\ge 1-\frac{\sigma^2}{\epsilon^2n}$$