Exponential Martingales

Let \(u\) be an integrable Borel function on \([0, 1)\) relative to the Lebesgue measure. For each \(n \ge 1\) and \(x \in [0, 1)\), let \(I_{n}(x) = [k2^{-n}, (k + 1)2^{-n})\) be the interval that contains \(x\) as \(k\) varies from 0 to \(2^{n} - 1\). Show that for Lebesgue-a.e. \(x\), \[ \lim _{n \to \infty } 2^{n}\int _{I_{n}(x)} u(y)\,dy = u(x). \]

(Hint: Cast this in a suitable martingale framework then use the result of problem 3.)

Let \((X_n)\) be i.i.d. uniform random variables on \([0,1]\). For each \(n\geq 1\), define \[ Y_n=n\min \{X_k:1\leq k\leq n\}. \] Prove that as \(n\to \infty \), \(Y_n\) converges weakly to an exponential random variable with mean one.

Suppose \((M_n)\) is a martingale with \(M_0=0\) and \(|M_{n+1}-M_n|\leq 1\) for all \(n\geq 0\).

1. Prove that there exists a finite positive constant \(C\) such that \(e^x\leq 1+x+Cx^2\) for all \(x\in [-1,1]\).
2. Prove that \(\mathbb {E}[e^{\lambda M_n}]\leq (1+C\lambda ^2)^n\leq e^{C\lambda ^2n}\) for all \(n\geq 1\) and \(\lambda \in (0,1)\).
3. Prove that \(M_n/n^\delta \to 0\) almost surely for every \(\delta >1/2\).

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: \(M_n=M_0+\sum _{j=1}^{n}(M_j-M_{j-1})\).

Suppose that \(X_1,X_2,\ldots \) are i.i.d. exponential random variables with parameter \(\lambda >0\). Prove that \[ \overline {\lim }_{n\to \infty }\left (\frac {X_n}{\log n}\right )=\lambda ^{-1} \qquad \text {and}\qquad \underline {\lim }_{n\to \infty }\left (\frac {\log X_n}{\log n}\right )=-1. \]

Let \(X_1,X_2,\ldots \) be i.i.d. with \(\mathbb {P}\{X_1=1\}=p\) and \(\mathbb {P}\{X_1=-1\}=q\), where \(p\notin \{0,\frac {1}{2},1\}\) and \(q=1-p\). Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\) and consider the stopping time, \(T:=\inf \{n\geq 1:|S_n|=5\}\), where \(\inf \emptyset :=\infty \). Prove that \(T\in L^1(\mathbb {P})\) and compute \(\mathbb {E}(T)\). (HINT: Begin by proving that \(M_n:=(q/p)^{S_n}\) and \(N_n:=S_n-(p-q)n\) are martingales.)

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. exponential random variables with mean one; that is, \(\mathbb {P}\{X_n>x\}=e^{-x}\) for all \(x>0\). Prove that \[ \mathbb {P}\left \{\limsup _{n\to \infty }\frac {X_n}{\log n}=1\right \}=1. \]

Let \(X_1,\ldots ,X_n\) be iid exponential(1) random variables. Find the joint pdf of the pair \((X_{(1)},X_{(3)})\), where \(X_{(i)}\) denotes the \(i\)th order statistic, i.e. \[ X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(n)}. \]

Let \(X_1,X_2\) be independent and identically distributed exponential\((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Find the joint density function of \[ Y_1=X_1+X_2 \quad \text {and}\quad Y_2=\frac {X_1}{X_1+X_2}. \]

Suppose \(\{X_j\}_{j=0}^{\infty }\) is a martingale such that \(X_0=0\) and \(|X_{i+1}-X_i|\leq 1\) for all \(i\geq 0\). Prove that \(\lim _{n\to \infty }n^{-\delta }X_n=0\), a.s. for every \(\delta >1/2\).

Suppose \(X_1,X_2,\ldots \) are independent, identically-distributed exponential random variables with mean \(\lambda >0\). Prove that \[ \max (X_1\ldots ,X_n)-\frac {1}{\lambda }\ln n\Rightarrow X, \] and compute \(\mathbb {P}\{X>x\}\) for all \(x>0\).

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \). Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\). For fixed \(n\geq 2\), define \[ M_1:=\frac {S_n}{n},\qquad M_2:=\frac {S_{n-1}}{n-1},\qquad \ldots \qquad M_n:=\frac {X_1}{1}. \] (a) Show that \(\mathbb {E}[X_k\mid S_n]=S_n/n\) for \(1\leq k\leq n\).

(b) Show that \(M_1,M_2,\ldots ,M_n\) is a martingale.

Let \(X\) be an exponential random variable with mean 1, that is, \(\mathbb {P}(X>u)=e^{-u}\) for all \(u>0\). Evaluate \(\mathbb {E}[X\mid X\wedge t]\) and \(\mathbb {E}[X\mid X\vee t]\) for all \(t>0\). Here \(a\wedge b:=\min (a,b)\) and \(a\vee b:=\max (a,b)\).

Let \(X_1,X_2,\ldots \) be random variables such that the partial sums \(S_n:=X_1+X_2+\cdots +X_n\) determine a martingale. Show that \(\mathbb {E}[X_iX_j]=0\) if \(i\neq j\).

Suppose \(X_1, \ldots , X_n\) is an i.i.d. sample from the exponential density with mean \(\theta \), where \(\theta > 0\) is unknown. Find an estimator of \(\theta \) based on \(X_1, \ldots , X_n\). Is your estimator complete? Is it sufficient? Justify your reasoning.

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 \(\{X_n\}_{n=1}^{\infty }\) 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 \(T\).

Let \(X_1\) and \(X_2\) be two independent, identically distributed exponential (1) random variables. Compute the joint density of \(Y_1=X_1+X_2\) and \(Y_2=X_1/(X_1+X_2)\).

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. standard normal random variables. Prove that \[ M_n:=(n+1)^{-1/2}\exp \left (\frac {S_n^2}{2n+2}\right ) \] defines a martingale, where \(S_n=\sum _{i=1}^{n}X_i\).

\(Z_1\) and \(Z_2\) are two independent identically distributed exponential \((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ \dfrac {1}{\lambda }e^{-t/\lambda }, & \text {if } 0\leq t<\infty . \end {cases} \] Let \(X=Z_1\) and \(Y=Z_1+Z_1Z_2\). Compute

(a) \(E(Y\mid X=x)\)

(b) \(\operatorname {var}(E(Y\mid X))\).

Let \(\{S_n,\mathcal {F}_n,n\geq 1\}\) be a nonnegative martingale with \(ES_n=1\). Show that for all \(\lambda >0\), \[ P\{S_n\geq \lambda ,\text { for some }n\geq 1\}\leq \frac {1}{\lambda }. \]

Let \(\{S_n,\mathcal F_n,\,n\ge 1\}\) be a nonnegative martingale with \(\mathbb {E} S_n=1\). Show that for all \(\lambda >0\), \[ \mathbb {P}\{S_n\ge \lambda ,\text { for some } n\ge 1\}\le \frac {1}{\lambda }. \]