Probability Qualifying Exam

Let \(X\), \(Y\), \(Z\) be random variables on some probability space. Recall that

is the conditional expectation of the indicator function of the event \(\{X \in A\}\), given the \(\sigma \)-algebra \(\sigma \{Y, Z\}\) generated by \(Y\) and \(Z\). We say that \(X\) and \(Y\) are independent, given \(Z\), if for all Borel sets \(A, B \subset \mathbb {R}\),

Show that this condition is equivalent to having \(P(X \in A \mid Y, Z) = P(X \in A \mid Z)\) \(P\)-almost surely, for all Borel sets \(A\).

Let \(Y_{k}\) be i.i.d., \(Y_{k} \ge 0\), and \(\mathbb {E}[Y_{k}] \le 1\) for all \(k\). Prove the existence of the almost sure limit \(M_{\infty } = \lim _{n \to \infty } \prod _{i=1}^{n} Y_{i}\) and describe this random variable explicitly. Do not forget the degenerate cases.

Let \(\{\mathcal {F}_{n}\}_{n \ge 0}\) be a filtration on the probability space \((\Omega , \mathcal {F}, P)\), let \(X \in L^{1}(\Omega , \mathcal {F}, P)\), and define \(M_{n} = \mathbb {E}[X \mid \mathcal {F}_{n}]\). Let \(\tau \) be a stopping time and recall the definition \[ \mathcal {F}_{\tau } = \{A \in \mathcal {F} : A \cap \{\tau = n\} \in \mathcal {F}_{n}\ \forall n \ge 0\}. \]

1. Show that \(1_{\{\tau < \infty \}}M_{\tau }\) is integrable.
2. Show that \(\mathbb {E}[X \mid \mathcal {F}_{\tau }] = 1_{\{\tau < \infty \}}M_{\tau } + 1_{\{\tau = \infty \}}X\). Do not forget the necessary measurability condition.

Answer the following questions.

1. State the definition of convergence in probability. Using only this definition, show that \(X_{n} \to X\) in probability if and only if \(\mathbb {E}[1 \wedge |X_{n} - X|] \to 0\). Do not appeal to any theorems that trivialize the problem.
2. State the definition of almost sure convergence. Prove that if \(X_{n} \to X\) almost surely, then \(X_{n} \to X\) in probability. Again, do not appeal to any theorems that trivialize the problem.

Let \(\{X_{n}\}_{n \ge 1}\) be i.i.d. with mean 0 and variance 1. Let \(\{a_{n}\}_{n \ge 1}\) be a sequence of nonzero real numbers and \[ A_{n} = \left (\sum _{k=1}^{n} a_{k}^{2}\right )^{1/2}. \] Assume that \[ A_{n}^{-1}\left (\max _{1 \le k \le n} |a_{k}|\right ) \to 0 \qquad \text {as } n \to \infty . \] Find (and prove) a distributional limit for \[ A_{n}^{-1}\sum _{k=1}^{n} a_{k}X_{k} \qquad \text {as } n \to \infty . \]

Let \(\{X_{n}\}_{n \ge 1}\) be i.i.d. Normal random variables with mean 0 and variance 1. Let \(S_{0} = 0\) and \(S_{n} = X_{1} + \cdots + X_{n}\) for \(n \ge 1\). Fix \(t \in \mathbb {R}\) and for \(n \ge 0\) let \[ M_{n} = e^{tS_{n} - t^{2}n/2}. \]

1. Prove that \(M_{n}\) is a mean one martingale relative to the filtration \(\mathcal {F}_{n} = \sigma (X_{1}, \ldots , X_{n})\).
2. Prove that if \(\max _{j \le n} S_{j} \ge nt\), then \(\max _{j \le n} M_{j} \ge e^{t^{2}n/2}\).
3. Conclude that \(P\{\max _{j \le n} S_{j} \ge nt\} \le e^{-t^{2}n/2}\) and then that for any \(c > \theta > 1\) we have with probability one: \(\exists k_{0} \ge 1\) such that for \(k \ge k_{0}\) \[ \frac {\max _{j \le \theta ^{k}} S_{j}} {\sqrt {2c\theta ^{k-1}\log \log \theta ^{k-1}}} \le 1. \]
4. Deduce that almost surely \[ \limsup _{n \to \infty } \frac {S_{n}}{\sqrt {2n\log \log n}} \le 1. \]