Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically distributed random variables with \(\xi_{n} \in \mathcal{L}^{2}(\mathscr{A})\) and \(\int \xi_{n} d \mathbb{P}=0\). Set \(\mathscr{A}_{n}:=\) \(\sigma\left(\xi_{1}, \xi_{2}, \ldots, \xi_{n}ight)\).

(i) Show, without using Example 23.3 (vi), that \(S_{n}^{2}:=\left(\xi_{1}+\xi_{2}+\cdots+\xi_{n}ight)^{2}\) is a submartingale w.r.t. \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\).

(ii) Show that there is a constant \(\kappa\) such that \(S_{n}^{2}-\kappa n\) is a martingale w.r.t. \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\).

Data from example 23.3 (vi)

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(iv) Let (un)neNo be a submartingale. Then (u)neN, is a submartingale. Indeed: take A E, and observe that {un >0} En. Then Un+1 du U+1 L La Lon 1 du > in{un>0} (23.2) > "Lantampay " - Sunt un d = u d.