Probability Prelim Exam 1.1 August 2019

Assume that \(A_i\), \(i=1,2,\ldots \), are independent events. Prove that \[ \frac {1}{n}\sum _{i=1}^{n}\left (\mathbf {1}_{A_i}-\mathbb {P}(A_i)\right ) \] converges to zero in probability as \(n\to \infty \), where where \(\mathbf {1}_{A_i}\) is the indicator function of the event \(A_i\).

Let \(Z_n\) be a martingale difference with respect to a filtration \(\mathcal {F}_n\), i.e. \(\mathbb {E}[Z_{n+1}\mid \mathcal {F}_n]=0\) for all \(n\geq 0\). Assume the following hold:

1. \(n^{-1/2}\max _{m\leq n}|Z_m|\to 0\) in probability.
2. \(n^{-1}\sum _{m=1}^{n}|Z_m|^2\to 1\) in probability.
3. \(\sum _{m=1}^{n}|Z_m|^2\leq 2n\) almost surely, for all \(n\).

Prove that \(M_n=n^{-1/2}\sum _{m=1}^{n}Z_m\) converges in distribution to a standard normal.

Hint: Prove that \(|1+ix|\leq e^{x^2}\) for all \(x\in \mathbb {R}\) and \(e^{ix}=(1+ix)e^{-x^2/2+r(x)}\) with \(|r(x)|\leq x^3\) for all \(x\in \mathbb {R}\). Let \[ T_n=\prod _{m=1}^{n}\left (1+itn^{-1/2}Z_m\right ) \] and \[ U_n= e^{-\frac {t^2}{2n}\sum _{m=1}^{n}Z_m^2 +\sum _{m=1}^{n}r(tn^{-1/2}Z_m)}. \] Compute \(\mathbb {E}[T_n]\) then prove that \(\mathbb {E}[T_nU_n]\to e^{-t^2/2}\) and conclude.

Let \(X_k\) have characteristic function \(\phi _k\) with \(X_1,X_2,\ldots \), independent random variables. Show that \(\sum _{k=1}^{n}X_k\) converges almost surely \(\Longleftrightarrow \) there exists a neighborhood \(U\) of 0 and a function \(h\) such that \[ \prod _{k=1}^{n}\phi _k(u) \xrightarrow [n\to \infty ]{} h(u)\ne 0, \] for all \(u\in U\). Hint: For the \(\Leftarrow \) direction consider the characteristic function of the partial sums \(\sum _{k=m}^{n}X_k\).

Let \(Z_1,Z_2,\ldots \) be iid with \(\mathbb {P}(Z=1)=\mathbb {P}(Z=-1)=1/2\), and let \(c_1,c_2,\ldots \) be given constants.

1. Express the characteristic function of \(\sum _{k=1}^{n}c_kZ_k\) in terms of standard elementary functions.
2. Show that \(\sum _{k\geq 1}c_kZ_k\) converges almost surely \(\Longleftrightarrow \) \(\sum _{k=1}^{\infty }c_k^2<\infty \). For this you can use results we proved in class, or other problems on this exam.

Let \((Z_n)\) be i.i.d. random variables with \(\mathbb {P}(Z_1=1)=\mathbb {P}(Z_1=2)=1/2\). Show that \(\mathbb {E}[Z_1\cdots Z_n]=(1.5)^n\) and yet \(Z_1\cdots Z_n/(1.45)^n\to 0\) almost surely. Identify the set of numbers \(a>0\) for which \(Z_1\cdots Z_n/a^n\to 0\) almost surely.

(Pólya’s Urn) An urn initially contains \(r\) red and \(b\) blue balls. A ball is chosen uniformly at random (i.e. with probability \(1/(r+b)\) each). If it comes up red (resp. blue), then it is returned and another red (resp. blue) ball is added to the urn. The process is repeated indefinitely. Let \(R_n\) be the number of red balls in the urn after \(n\) draws.

1. A vector \((X_1,X_2,\ldots ,X_n)\) of random variables is said to be exchangeable if \[ (X_1,\ldots ,X_n)\sim (X_{\pi (1)},\ldots ,X_{\pi (n)}) \qquad \text {for all permutations }\pi , \] where \(\sim \) denotes equality in distribution. If \(X_i\) is the indicator that a red ball is drawn from the urn at time \(i\), prove that \((X_1,\ldots ,X_n)\) is exchangeable.
2. Find the mean and variance of \(S_n=X_1+\cdots +X_n\), the total number of red balls added to the urn up to time \(n\).

Let \(\{X_{i,n}:i,n\geq 1\}\) be i.i.d. random variables with mass function \(\{f(x):x=0,1,2,\ldots \}\). Fix \(z\in \{1,2,\ldots \}\). Define random variables \((Z_n)\) as follows: \[ Z_0=z \qquad \text {and}\qquad Z_n=\sum _{i=1}^{Z_{n-1}}X_{i,n} \qquad \text {for } n\geq 1. \] I.e. \(Z_n\) is a branching process with offspring distribution \(f\) and initial population \(z\). For \(s\in [0,1]\) let \(g(s)=\sum _{x=0}^{\infty }f(x)s^x\). Suppose that \(s_0\in (0,1)\) solves \(g(s)=s\). Show that then \(s_0^{Z_n}\) is a martingale in the filtration \(\mathcal {F}_n=\sigma (X_{i,m}:i\geq 1,m\leq n)\). Use this to conclude that \[ \mathbb {P}(Z_n=0\text { for some }n\geq 0)=s_0^z. \]

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables, and let \(Z=X+Y\).

1. Show that \(\mathbb {E}[Z\mid X>0,Y>0]=2\sqrt {2/\pi }\).
2. Find the distribution and the density of \(Z\) given that \(X>0\) and \(Y>0\). (You can use the CDF of a standard normal, \(\Phi (x)=\mathbb {P}(X\leq x)\), in your expression.)

Suppose \(X_1,X_2,\ldots \) are iid with \(\mathbb {E}|X_i|=\infty \). Let \(S_n=X_1+\cdots +X_n\), and let \(A\) be the event that \(M_n=S_n/n\) converges to a finite limit. Let \(B\) be the event that \(|X_n|\geq n\) infinitely often.

1. State the definition of \(B\) in terms of unions and intersections.
2. Show that \(\mathbb {P}(B)=1\).
3. Verify the identity \[ M_n-M_{n+1}=\frac {M_n}{n+1}-\frac {X_{n+1}}{n+1} \] and use it to show that \(A\cap B=\emptyset \).
4. Use the above to prove that \(\mathbb {P}(A)=0\).

Let \(X_1,X_2,\ldots \) be independent random variables with \[ X_n= \begin {cases} 1, & \text {with probability }1/(2n),\\ 0, & \text {with probability }1-1/n,\\ -1, & \text {with probability }1/(2n). \end {cases} \] Let \(Y_1=X_1\) and for \(n\geq 2\) define \[ Y_n= \begin {cases} X_n, & \text {if }Y_{n-1}=0,\\ nY_{n-1}|X_n|, & \text {if }Y_{n-1}\ne 0. \end {cases} \] Show that \(Y_n\) is a martingale with respect to \(\mathcal {F}_n=\sigma (Y_1,\ldots ,Y_n)\). Show that \(Y_n\) does not converge almost surely. Why does the martingale convergence theorem not apply?