Probability Qualifying Exam

Let \(q\) be an integrable random variable. Prove that for each \(\epsilon > 0\) we can pick \(\delta > 0\) so that \(\mu (A) < \delta \) implies \[ \int _{A}|q|\,d\mu < \epsilon . \] Let \(f \ge 0\). (Hint: Show first that for any \(f \ge 0\), \(\int f \wedge n\,d\mu \nearrow \int f\,d\mu \) as \(n \to \infty \).)

At time \(n = 0\), an urn contains \(B_{0} = 1\) blue, and \(R_{0} = 1\) red ball. At each time \(n = 1, 2, 3, \ldots \), a ball is chosen at random from the urn and returned to the urn, together with a new ball of the same color. Let \(B_{n}\) and \(R_{n}\) be the number of blue, respectively, red balls in the urn after the \(n\)-th turn of this procedure. (Note that

Denote by \(\mathcal {F}_{n} = \sigma (B_{j}, 0 \le j \le n) = \sigma (R_{j}, 0 \le j \le n)\), \(n \ge 0\), the

natural filtration of the process. Let

be the proportion of blue balls in the urn just after time \(n\).

1. Show that \((M_{n})_{n \ge 0}\) is a martingale in the filtration \((\mathcal {F}_{n})_{n \ge 0}\).
2. Show that \(P(B_{n} = k) = 1/(n + 1)\) for all integers \(k \in [1, n + 1]\). (Hint: Write down the probability of choosing \(k\) blue and \(n - k\) red balls in whatever fixed order [hence, ending up with \(k + 1\) blue and \(n - k + 1\) red balls].)
3. Show that \(M_{\infty } = \lim _{n \to \infty } M_{n}\) exists almost surely and compute its distribution. (Hint: use part (b).)

Let \((a_{n})_{n \ge 1}\) be a nondecreasing sequence of positive numbers such that \[ \lim _{n \to \infty }\frac {a_{n}^{2}}{\sum _{k=1}^{n}a_{k}^{2}} = 0. \]

Let \((X_{n})_{n \ge 1}\) be a sequence of i.i.d. random variables with mean 0 and variance 1. Prove that \[ \frac {\sum _{k=1}^{n} a_{k}X_{k}}{\sqrt {\sum _{k=1}^{n}a_{k}^{2}}} \]

converges weakly to the standard normal distribution.

Let \(A_{n}\) be a sequence of independent events with \(P(A_{n}) < 1\) for all \(n\). Show that \[ P\left (\bigcup _{n} A_{n}\right ) = 1 \quad \text {implies} \quad \sum _{n} P(A_{n}) = \infty . \]

Let \(X_{1}, X_{2}, \ldots \) be i.i.d. with characteristic function \(\phi \).

1. Show that if \(\phi '(0) = ia\) and \(S_{n} = X_{1} + \cdots + X_{n}\), then \(S_{n}/n \to a\) in probability. (Hint: first show that if \(c_{n} \to c \in \mathbb {C}\), then \((1 + c_{n}/n)^{n} \to e^{c}\).)
2. Show that if \(S_{n}/n \to a\) in probability, then \(\phi (t/n)^{n} \to e^{iat}\) as \(n \to \infty \) through the integers.

Let \(Y_{1}, Y_{2}, \ldots \) be nonnegative i.i.d. random variables with \(\mathbb {E}[Y_{m}] = 1\) and \(P(Y_{m} = 1) <\) 1.

1. Show that \(X_{n} = \prod _{m \le n}Y_{m}\) defines a martingale.
2. Show that \(X_{n} \to 0\) a.s. (Hint: note that \(|X_{n+1} - X_{n}| = X_{n}|Y_{n+1} - 1|\).)
3. Use the strong law of large numbers to conclude \(\frac {1}{n}\log (X_{n}) \to c < 0\).