Question

1) a) Let {X_n} be a sequence of uniformly bounded random variables such that n^{-2}S_{n^2} xrightarrow{a.s.} 0, where S_n = sum_{i=1}^n X_i. Then show that n^{-1}S_n xrightarrow{a.s.} 0. b) For any sequence of i.i.d random variables {X_n} with mean zero and variance 16, calculate lim_{n o infty} mathbb{E}[frac{|S_n|}{sqrt{n}}], where S_n = sum_{i=1}^n X_i.

          1) a) Let {X_n} be a sequence of uniformly bounded random variables such that
n^{-2}S_{n^2} xrightarrow{a.s.} 0,
where S_n = sum_{i=1}^n X_i. Then show that n^{-1}S_n xrightarrow{a.s.} 0.
b) For any sequence of i.i.d random variables {X_n} with mean zero and variance 16, calculate lim_{n 	o infty} mathbb{E}[frac{|S_n|}{sqrt{n}}], where S_n = sum_{i=1}^n X_i.

$1) a) Let Xn be a sequence of uniformly bounded random variables such that n^-2Sn^2 xrightarrowa.s. 0, where Sn = sumi=1^n Xi. Then show that n^-1Sn xrightarrowa.s. 0. b) For any sequence of i.i.d random variables Xn with mean zero and variance 16, calculate limn o infty mathbbE[frac|Sn|sqrtn], where Sn = sumi=1^n Xi.$

Added by Brianna G.

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