Probability Qualifying Exam

Suppose \(\{r_{k}\}_{k \in \mathbb {Z}_{+}}\) is a sequence of real numbers such that \(n^{-1}\sum _{k=1}^{n}|r_{k}| \to 0\) as \(k \to \infty \). Let \(\{X_{k}\}_{k \in \mathbb {N}}\) be a sequence of \(L^{2}\) random variables such that \(\mathbb {E}[X_{k}] = 0\) and \(\mathbb {E}[X_{k}X_{\ell }] \le \) \(r_{\ell -k}\) for all \(k \le \ell \). Let \(S_{n} = X_{1} + \cdots + X_{n}\). Show that \(n^{-1}S_{n}\) converges in \(L^{2}\) as \(n \to \infty \) and identify the limit.

Answer the following two questions.

1. Let \(\{X_{n}\}_{n \in \mathbb {N}}\) be independent, almost surely finite random variables. Show that \[ \sup _{n} X_{n} < \infty \text { a.s.} \] if, and only if, \(\exists c < \infty \) such that \[ \sum _{n} P(X_{n} > c) < \infty . \]
2. Let \(\{A_{n}\}_{n \in \mathbb {N}}\) be a sequence of events with \(P(A_{n}) \ge \delta \) for each \(n\). Show that \[ P(A_{n}\text { happens infinitely often}) \ge \delta . \]

Consider a probability space \((\Omega , \mathcal {F}, P)\). Let \(X \in L^{1}\). Suppose \((\mathcal {F}_{n})_{n \in \mathbb {N}}\) is an increasing sequence of \(\sigma \)-algebras such that \(\cup _{n \in \mathbb {N}}\mathcal {F}_{n}\) generates \(\mathcal {F}\). Prove the Doob martingale convergence theorem that states that \(\mathbb {E}[X \mid \mathcal {F}_{n}]\) converges almost surely to \(X\).

Let \(u\) be an integrable Borel function on \([0, 1)\) relative to the Lebesgue measure. For each \(n \ge 1\) and \(x \in [0, 1)\), let \(I_{n}(x) = [k2^{-n}, (k + 1)2^{-n})\) be the interval that contains \(x\) as \(k\) varies from 0 to \(2^{n} - 1\). Show that for Lebesgue-a.e. \(x\), \[ \lim _{n \to \infty } 2^{n}\int _{I_{n}(x)} u(y)\,dy = u(x). \]

(Hint: Cast this in a suitable martingale framework then use the result of problem 3.)

Fix an integer \(n \ge 1\). Let \(\{Y_{k}\}_{k \in \mathbb {N}}\) be i.i.d. uniform on \(\{1, 2, \ldots , n\}\). Let

be the first time we get a repeated sample. Find an exponent \(\gamma \) and a nondegenerate distribution \(\mu \) such that, as \(n \to \infty \), \(n^{-\gamma }\sigma _{n}\) converges weakly to the distribution \(\mu \). (Hint: look at tail probabilities \(P(n^{-\gamma }\sigma _{n} \ge x)\).)

Give an example of a sequence of independent random variables \(\{Z_{k}\}_{k \in \mathbb {N}}\) satisfying \(\mathbb {E}[Z_{k}] = 0\), \(\operatorname {Var}(Z_{k}) = 1\), for each \(k\), and such that \((Z_{1} + \cdots + Z_{n})/\sqrt {n}\) does not converge in distribution to a standard normal. Justify any claims you make.

Let \(\{X_{n}\}_{n \in \mathbb {N}}\) be independent random variables such that for each \(n\), \(X_{n}\) takes the values \(+1\) and \(-1\) with probability \((1 - 2^{-n})/2\) each and the value \(2^{k}\) with probability \(2^{-k}\), for integers \(k > n\).

1. Show that \(\mathbb {E}[|X_{n}|] = \infty \) for all \(n \in \mathbb {N}\).
2. Show that \((X_{1} + \cdots + X_{n})/\sqrt {n}\) converges in distribution to a standard normal. (Hint: Check that we can write \(X_{n} = (1 - B_{n})Y_{n} + 2^{n}B_{n}Z_{n}\), where \(B_{n}\) is \(\operatorname {Bernoulli}(2^{-n})\), \(Y_{n}\) takes values \(\pm 1\) equally likely, \(Z_{n}\) takes the value \(2^{k}\) with probability \(2^{-k}\), for \(k \in \mathbb {N}\), and \(\{B_{n}, Z_{n}, Y_{n} : n \ge 1\}\) are mutually independent. Now, \(X_{n}\) is a small perturbation of \((1 - B_{n})Y_{n}\). Work out the details.)