Probability Qualifying Exam

Without using the Central Limit Theorem or the Law of the Iterated Logarithm, prove the Weak Law of Large Numbers when \(X_{i} \in L^{1}\), i.e. show that when \(X_{1}, X_{2}, \ldots \) is a sequence of i.i.d. variables with \(\mathbb {E}[X_{1}] = \mu \) and \(X_{1} \in L^{1}\), then

Hint: first, prove the claim when \(X_{1} \in L^{2}\). Then for \(a > 0\), consider

Let \(\{A_{i}\}_{i=1}^{\infty }\) be a sequence of independent events such that \(\sum _{i=1}^{\infty } P(A_{i}) = \infty \). Prove that the \(A_{n}\) occur infinitely often with probability 1 (the converse of the Borel-Cantelli Lemma), i.e. that \(\sum _{i=1}^{\infty }\mathbf {1}_{A_{i}} = \infty \) almost surely.

Show that if \(X_{n} \to X\) in probability, then \(X_{n} \Rightarrow X,\) i.e. \(X_{n}\) converges weakly to \(X\).

Recall the inversion formula: if \(\mu \) is a probability measure and \(\widehat {\mu }\) is its characteristic function, then for all \(a < b\):

1. Prove that if \(\widehat {\mu } \in L^{1}\), then \(\mu (\{a\}) = 0\) for all \(a \in \mathbb {R}\) and \(\mu \) has a probability density function given by

   \[ f(y) = \frac {1}{2\pi }\int _{-\infty }^{\infty } e^{-ity}\widehat {\mu }(t)\,dt. \]

2. Show that if \(X_{1}, \ldots , X_{n}\) are independent and uniformly distributed on \((-1, 1)\), then for \(n \ge 2\), \(X_{1} + \cdots + X_{n}\) has density

   \[ f(y) = \frac {1}{\pi }\int _{0}^{\infty } \left (\frac {\sin t}{t}\right )^{n}\cos (ty)\,dt. \]

Prove that if \(\{X_{i}\}_{i=1}^{\infty }\) are i.i.d. Uniform-\([0,1]\) random variables, then \[ \frac {4\sum _{i=1}^{n} iX_{i} - n^{2}}{n^{3/2}} \] converges weakly, and identify the limiting distribution. (Recall \(\sum _{i=1}^{n} i = \frac {n(n+1)}{2}\) and \(\sum _{i=1}^{n} i^{2} = \frac {n(n+1)(2n+1)}{6}\).)

Let \(X_{n}, Y_{n}\) be positive, in \(L^{1}\), and measurable with respect to the filtration \(\mathcal {F}_{n}\). Suppose

with \(\sum Y_{n} < \infty \) a.s.

1. Show that \(\prod _{i=1}^{n}(1 + Y_{i})\) converges a.s. to a finite limit.

2. Show

   \[ M_{n} = \frac {X_{n}}{\prod _{i=0}^{n-1}(1 + Y_{i})} \]
   is a super-martingale.
3. Use (a) and (b) to prove that \(X_{n}\) converges a.s. to a finite limit.