Ph.D. Qualification Examination in Probability January 2005

Prove that for all submartingales \(\{X_n\}_{n=1}^{\infty }\), \[ \mathbb {P}\left \{\max _{n\le m}|X_n|\ge \lambda \right \} \le \frac {3}{\lambda }\max _{n\le m}\mathbb {E}\{|X_n|\}, \qquad \lambda >0,\; m=1,2,\ldots . \]

Suppose \(X_1,X_2,\ldots \) are independent with distribution, \[ \mathbb {P}\{X_n=n^\alpha \}=\mathbb {P}\{X_n=-n^{-\alpha }\}=\frac {1}{2}, \qquad n=1,2,\ldots . \] Prove that \(\sum _{i=1}^{n}X_n/n\to 0\) a.s. if \(0<\alpha <\frac {1}{2}\).

Suppose \(X_{\alpha ,\beta }\) has a Gamma distribution with parameters \((\alpha ,\beta )\). That is, the density function of \(X_{\alpha ,\beta }\) is \[ f(x)=\frac {\beta ^\alpha }{\Gamma (\alpha )}x^{\alpha -1}e^{-\beta x}, \qquad x>0, \] where \(\Gamma (\alpha )=\int _0^\infty x^{\alpha -1}e^{-x}\,dx\) for all \(\alpha >0\). Prove that \((\beta X_{\alpha ,\beta }-\alpha )/\sqrt {\alpha }\Rightarrow N(0,1)\).

Let \(X\) be a mean-zero random variable with \(\mathbb {E}(X^2)=\sigma ^2<\infty \). Prove Cantelli’s inequality: For all \(\lambda >0\), \[ \mathbb {P}\{X>\lambda \}\le \inf _{t>0}\left [\frac {t^2+\sigma ^2}{(t+\lambda )^2}\right ] =\frac {\sigma ^2}{\sigma ^2+\lambda ^2}. \] Prove that this is a better bound than Chebyshev’s inequality.

If \(X\in L^2(\Omega ,\mathcal {F},\mathbb {P})\), then prove that for all sub-\(\sigma \)-algebras \(\mathcal {G}\) of \(\mathcal {F}\), \[ \operatorname {Var}(X)=\mathbb {E}[\operatorname {Var}(X\mid \mathcal {F})]+\operatorname {Var}(\mathbb {E}\{X\mid \mathcal {F}\}) \qquad \text {a.s.} \]

Suppose \(Y,X_1,X_2,\ldots \) have the following properties:

(a) \(Y\) has the exponential distribution. That is, \(\mathbb {P}\{Y>t\}=e^{-t}\) for \(t>0\).

(b) Conditionally on \(Y\), \(X_1,\ldots ,X_n\) are i.i.d. exponentials with parameter \(Y\), i.e., for all \(t_1,\ldots ,t_n>0\), \[ \mathbb {P}(X_1>t_1,\ldots ,X_n>t_n\mid Y) =\prod _{i=1}^{n}e^{-Yt_i} =e^{-Y\sum _{i=1}^{n}t_i} \qquad \text {a.s.} \]

Compute \(\mathbb {E}[Y\mid X_1,\ldots ,X_n]\). Use this to prove that \(\{M_n\}_{n=1}^{\infty }\) is a martingale, where \[ M_n=\frac {n+1}{1+X_1+\cdots +X_n} \qquad n=1,2,\ldots . \]

Suppose \(X_1,X_2,\ldots \) have common mean \(\mu \) and variances \(\sigma _1^2,\sigma _2^2,\ldots \). Prove that if \(\sup _n\sigma _n^2<\infty \) and \(\lim _{|i-j|\to \infty }\mathbb {E}[X_iX_j]=0\), then \((X_1+\cdots +X_n)/n\) converges to \(\mu \) in probability.

Prove that whenever \(X_n\Rightarrow X\) and \(Y_n\to c\) in probability for a non-random \(c>0\), then \(X_n/Y_n\Rightarrow c^{-1}X\). Use this to prove the following: If \(\xi _1,\xi _2,\ldots \) are i.i.d. with \(\mathbb {E}[\xi _1]=\mu \) and \(\operatorname {Var}(\xi _1)=\sigma ^2\) then \[ \frac {S_n-n\mu }{\sqrt {\sum _{i=1}^{n}(\xi _i-\bar {\xi }_n)^2}} \Rightarrow N(0,1). \] where \(S_n=\xi _1+\cdots +\xi _n\), and \(\bar {\xi }_n=S_n/n\).

Suppose \(\{X_n\}_{n=1}^{\infty }\) is a martingale that is bounded in \(L^1(\mathbb {P})\); i.e., \(\sup _n\|X_n\|_1<\infty \). Use Doob’s decomposition to prove that \(X_T\in L^1(\mathbb {P})\) for all stopping times \(T\).

Suppose \(\{X_n\}_{n=1}^{\infty }\) is a sequence of random variables that has the property that \(\sup _n|X_n|\le 1\) a.s. Then use Doob’s decomposition to prove that \(\sum _n X_n\) converges a.s. iff \(\sum _n\mathbb {E}[X_n\mid X_1,\ldots ,X_{n-1}]\) converges a.s.