Probability Prelim Exam 1.1 August 2017

Let \(X_1,X_2,\ldots \) be independent random variables, and suppose that \(p_i:=\mathbb {P}\{X_i=0\}\) satisfies \(0<p_i<1\) for every \(i=1,2,\ldots \). For every \(n\geq 1\) let \[ N_n:=\sum _{i=1}^{n}\mathbf {1}_{\{X_i=0\}} \] denote the number of times the finite random sequence \(X_1,\ldots ,X_n\) enters zero. Prove that \[ \frac {N_n}{\sum _{i=1}^{n}p_i}\to 1 \qquad \text {in probability as }n\to \infty . \]

Let \(X_n\) be i.i.d. random variables with mean \(\mu \) and variance \(\sigma ^2<\infty \). Prove (without using the law of the iterated logarithm) that we have almost surely \[ \overline {\lim }_{n\to \infty }\frac {S_n-n\mu }{\sigma \sqrt {n}}=\infty \qquad \text {and}\qquad \underline {\lim }_{n\to \infty }\frac {S_n-n\mu }{\sigma \sqrt {n}}=-\infty . \]

Suppose \(X\) and \(Y\) are two independent, real-valued random variables, and suppose that \(\mathbb {P}\{X\in A\}=0\) for all Borel sets \(A\) of zero Lebesgue measure. Prove, carefully, that there exists a measurable function \(f:\mathbb {R}\to \mathbb {R}\) such that \[ \mathbb {P}\{X+Y\in B\}=\int _B f(x)\,dx \qquad \text {for all measurable sets }B\subseteq \mathbb {R}. \]

Let \(X_1,X_2,\ldots \) be a collection of i.i.d. random variables, taking values \(\pm 1\) with probability \(1/2\) each. Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\), \(T_0:=0\), and then iteratively define \[ T_{k+1}:=\inf \{n>T_k:S_n=0\} \qquad [\inf \emptyset :=\infty ]\quad \text {for all }k\geq 0. \] That is, \(T_k\) denotes the \(k\)th return time to zero. Prove that \(\lim _{k\to \infty }(T_k/k)=\infty \) a.s.

Suppose \(X\) is a nonnegative random variable that satisfies the following: There exist two real numbers \(a,b>0\) such that \(a^n\leq \mathbb {E}[X^n]\leq b^n\) for every integer \(n\geq 1\). Prove that \(\mathbb {P}\{a\leq X\leq b\}=1\).

Let \(X_1,\ldots ,X_m\) be \(m\geq 2\) i.i.d. standard normal random variables, and define the random vector \[ \mathbf {X}:= \begin {bmatrix} X_1\\ \vdots \\ X_m \end {bmatrix}. \] Prove that the distribution of \(\mathbf {X}\) is the same as the distribution of \(A\mathbf {X}\) for every \(m\times m\) orthogonal matrix \(A\) that is non random.

Suppose \(X\) is a discrete, non-negative random variable with mass function \(p\). Prove that \[ \lim _{n\to \infty }\left (\mathbb {E}[X^n]\right )^{1/n} = \sup \{x\in \mathbb {R}:p(x)>0\}. \]

Fix \(0<\lambda <\rho \) and let \(X\), \(Y\), and \(Z\) be three independent Gamma distributed random variables with common scale parameter 1 and shape parameters \(\lambda \), \(\rho -\lambda \), and \(\rho \), respectively. (A Gamma random variable with scale parameter 1 and shape parameter \(\lambda \) has pdf \(\frac {1}{\Gamma (\lambda )}x^{\lambda -1}e^{-x}\mathbf {1}_{\{x>0\}}\).) Let \[ U=\frac {ZX}{X+Y},\qquad V=\frac {ZY}{X+Y},\qquad \text {and}\qquad W=X+Y. \] Prove that random vector \((U,V,W)\) has the same distribution as \((X,Y,Z)\). (It may be useful to note that \(Z=U+V\).)

Let \(M_n\) be an \(L^1\)-bounded martingale with respect to a filtration \(\mathcal {F}_n\). Prove that there exist two non-negative martingales \(Y_n\) and \(Z_n\) (in the same filtration) such that \(M_n=Y_n-Z_n\). Hint: Use \(Z_{n,j}=\mathbb {E}[|M_j|\mid \mathcal {F}_n]\) for \(j\geq n\).

Let \(M_n\) be a martingale such that \(\sup _n\mathbb {E}[M_n^2]<\infty \). Prove that there exists a random variable \(M_{\infty }\in L^2\) such that \(M_n\to M_{\infty }\) in \(L^2\). Hint: \(M_n=M_0+\sum _{j=1}^{n}(M_j-M_{j-1})\).