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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 6 / c / ii

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Oxford 2017 Prelims Mathematics III 6 c
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ii
By Chebyshev's inequality and part (i),
0≤P(nSn​​>1+ϵ)≤P(​nSn​​−1​≥ϵ)≤n2ϵ2Var(Sn​)​=n2ϵ25n−2​⟶0.
(1)
This actually proves the stronger statement Sn​/nP​1​, namely convergence in probability, despite the statistical dependence between adjacent products.

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