 L2 convergence of averages of independent variables with bounded second moments

ID: l2-convergence-of-averages-of-independent-variables-with-bounded-second-moments

sup_{n} E ∣ X_{n} ∣^{2} < \infty

, write

Y_{N} = \frac{1}{N} \sum_{k = 1}^{N} X_{k}, m_{N} = \frac{1}{N} \sum_{k = 1}^{N} E X_{k} .

Then

∥ Y_{N} - m_{N} ∥_{2}^{2} = N^{- 2} \sum_{k \leq N} Var (X_{k}) = O (N^{- 1})

. Hence

Y_{N}

converges in

L^{2}

m_{N}

converge.

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