91Ó°ÊÓ

Let \(S_{n}\) be an asymmetric simple random walk with \(p>1 / 2\), and let \(\sigma^{2}=1-(p-q)^{2}\). Use the fact that \(X_{n}=\left(S_{n}-(p-q) n\right)^{2}-\sigma^{2} n\) is a martingale to show \(\operatorname{var}\left(T_{1}\right)=\left(1-(p-q)^{2}\right) /(p-q)^{3} \ldots\)

Show that when \(E|X|, E|Y|\), and \(E|X Y|\) are finite, each statement implies the next one and give examples with \(X, Y \in\\{-1,0,1\\}\) a.s. that show the reverse implications are false: (i) \(X\) and \(Y\) are independent, (ii) \(E(Y \mid X)=E Y\), (iii) \(E(X Y)=E X E Y\)

Let \(X_{n}\) and \(Y_{n}\) be positive integrable and adapted to \(\mathcal{F}_{n}\). Suppose $$ E\left(X_{n+1} \mid \mathcal{F}_{n}\right) \leq\left(1+Y_{n}\right) X_{n} $$ with \(\sum Y_{n}<\infty\) a.s. Prove that \(X_{n}\) converges a.s. to a finite limit by finding a closely related supermartingale to which (2.11) can be applied.

Use regular conditional probability to get the conditional Hölder inequality from the unconditional one, i.e., show that if \(p, q \in(1, \infty)\) with \(1 / p+1 / q=1\) then $$ E(\mid X Y \| \mathcal{G}) \leq E\left(|X|^{p} \mid \mathcal{G}\right)^{1 / p} E\left(|Y|^{q} \mid \mathcal{g}\right)^{1 / q} $$ Unfortunately, r.c.d.'s do not always exist. The first example was due to Dieudonné (1948). See Doob (1953), p. 624 , or Faden (1985) for more recent developments. Without going into the details of the example, it is easy to see the source of the problem. If \(A_{1}, A_{2}, \ldots\) are disjoint, then (1.1a) and (1.1c) imply $$ P\left(X \in U_{n} A_{n} \mid G\right)=\sum_{n} P\left(X \in A_{n} \mid \mathcal{G}\right) \text { a.s. } $$