Prelim in Probability, Spring 2016

If \(X\in L^2\), then it is easy to show that \(\mu \) is its mean if and only if \(\mathbb {E}[(X-a)^2]\) is uniquely minimized at \(a=\mu \). Assuming \(X\in L^1\), show that \(m\) is a median of \(X\) if and only if \(\mathbb {E}[|X-a|]\) is minimized (not necessarily uniquely) at \(a=m\).

(a) Find a simple asymptotic formula for \(\mathbb {P}(X>x)\), where \(X\) has the GAMMA\((\theta ,\alpha )\) density, \[ f(x)=\frac {1}{\theta ^{\alpha }\Gamma (\alpha )} x^{\alpha -1}e^{-x/\theta }, \qquad x>0. \]

(b) Let \(X_1,X_2,\ldots \) be i.i.d. GAMMA\((\theta ,\alpha )\). Find an increasing sequence \(a_n\) such that \(\limsup _{n\to \infty }X_n/a_n=1\) a.s., and prove it.

(a) Find the distribution of \(\int _0^1 B_t\,dt\), where \((B_t)\) is Brownian motion.

(b) Given an i.i.d. sequence \(X_1,X_2,\ldots \) with mean 0 and variance 1, what does Donsker’s invariance principle tell us about this sequence in connection with the distribution of \(\int _0^1 B_t\,dt\)?

Let \(X_1,X_2,\ldots \) be i.i.d. with common distribution \(\mathbb {P}(X_1=1)=\mathbb {P}(X_1=-1)=1/2\), and put \(S_0:=0\) and \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Consider the stopping time \(N:=\min \{n\geq 1:S_n=1\}\).

(a) Let \(g(u):=\mathbb {E}[u^{X_1}]\) for all \(u>1\), and show that the optional stopping theorem applies to the martingale \(M_n:=u^{S_n}g(u)^{-n}\), where \(u>1\), and the stopping time \(N\).

(b) Use part (a) to show that \[ \mathbb {E}[v^N]=\frac {1-\sqrt {1-v^2}}{v}, \qquad 0<v<1. \]

(c) Use part (b) to show that \[ \mathbb {P}(N=2m+1)= \frac {1}{m+1}\binom {2m}{m}\frac {1}{2^{2m+1}}, \qquad m\geq 0. \]

Suppose \(g\) and \(h\) are continuous on \(\mathbf {R}\) with \(g>0\) and \(|h(x)|/g(x)\to 0\) as \(|x|\to \infty \). If \(X_n\Rightarrow X\) and \(\sup _n\mathbb {E}[g(X_n)]<\infty \), show that \(\mathbb {E}[h(X_n)]\to \mathbb {E}[h(X)]\).

Let \(U_0,U_1,\ldots \) be independent and identically distributed random variables, uniform on \([0,1]\). Let \(N\) be a Poisson random variable with \(\mathbb {E}[N]=1\), independent of \(U_0,U_1,\ldots \). Compute \(\mathbb {E}[Y]\), where \(Y=\max _{0\leq k\leq N} U_k\).

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables uniform on \([0,1]\) and let \(\alpha >0\). Show that there are numerical sequences \(a_n\) and \(b_n\) such that \[ Y_n=\frac {\sum _{k=1}^{n}k^{\alpha }X_k-a_n}{b_n} \] converges in distribution to a standard normal random variable.

Let \(\{e_i,-\infty <i<\infty \}\) be independent and identically distributed random variables with \(\mathbb {E}[e_1]=0\) and \(\mathbb {E}[e_1^2]=\sigma ^2\) and \(\rho \) be a real number satisfying \(|\rho |<1\). Show that \[ Y=\sum _{i=0}^{\infty }\rho ^i e_i \] is finite with probability one. Compute \(\mathbb {E}[Y]\) and \(\mathbb {E}[Y^2]\). (You need to justify your steps when you compute these two expected values.)

Compute \[ \begin {aligned} \lim _{n\to \infty }(b-a)^{-n} \int _a^b\int _a^b\cdots \int _a^b \frac {x_1+x_2+\cdots +x_n}{x_1^2+x_2^2+\cdots +x_n^2} \,dx_1\,dx_2\cdots dx_n \end {aligned} \] if \(0<a<b<\infty \).

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with characteristic function \(\varphi \). Let \(N\) be a random variable with \[ \mathbb {P}(N=k)=\frac {1}{2^k}, \qquad k=1,2,\ldots . \] We assume that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent.

(a) Compute the characteristic function of \(Y=X_1+X_2+\cdots +X_N\).

(b) Can you weaken the condition that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent and the formula obtained in (a) remains valid?