Martingales and martingale convergence

Let $X$ be a martingale. Use the tower property for conditional expectation to deduce that \[\operatorname{\mathbb{E}}\left[X_{n+k}|\mathcal{F}_{n}\right]= X_n\,, \quad k=0,1,2,\dots\,.\]
Recall Thackeray’s martingale: let $Y_1,Y_2,\dots$ be a sequence of independent and identically distributed random variables, with $\operatorname{\mathbb{P}}\left(Y_1 = 1\right) = \operatorname{\mathbb{P}}\left(Y_1 = -1\right) = 1/2$. Define the Markov chain $M$ by \[ M_0 = 0; \qquad M_n = \begin{cases} 1-2^{n} & \quad\text{if $Y_1=Y_2 = \dots = Y_n = -1$,} \\ 1 &\quad\text{otherwise.}\end{cases}\]
1. Compute $\operatorname{\mathbb{E}}\left[M_n\right]$ from first principles.
2. What should be the value of $\operatorname{\mathbb{E}}\left[\widetilde{M}_n\right]$ if $\widetilde{M}$ is computed as for $M$ but stopping play if $M$ hits level $1-2^N$?
Consider a branching process $Y$, where $Y_0=1$ and $Y_{n+1}$ is the sum $Z_{n+1,1}+\ldots+Z_{n+1,Y_n}$ of $Y_n$ independent copies of a non-negative integer-valued family-size r.v. $Z$.
1. Suppose $\operatorname{\mathbb{E}}\left[Z\right]=\mu<\infty$. Show that $X_n=Y_n/\mu^n$ is a martingale.
2. Show that $Y$ is itself a supermartingale if $\mu < 1$ and a submartingale if $\mu > 1$.
3. Suppose $\operatorname{\mathbb{E}}\left[s^{Z}\right]=G(s)$. Let $\eta$ be the smallest non-negative root of the equation $G(s)=s$. Show that $\eta^{Y_n}$ defines a martingale.
4. Let $H_n=Y_0+\ldots+Y_n$ be the total of all populations up to time $n$. Show that $s^{H_n}/(G(s)^{H_{n-1}})$ is a martingale.
5. How should these three expressions be altered if $Y_0 = k \geq 1$?
Consider asymmetric simple random walk, stopped when it first returns to $0$. Show that this is a supermartingale if jumps have non-positive expectation, a submartingale if jumps have non-negative expectation (and therefore a martingale if jumps have zero expectation).
Consider Thackeray’s martingale based on asymmetric random walk. Show that this is a supermartingale or submartingale depending on whether jumps have negative or positive expectation.
Show, using the conditional form of Jensen’s inequality, that if $X$ is a martingale then $|X|$ is a submartingale.
A shuffled pack of cards contains $b$ black and $r$ red cards. The pack is placed face down, and cards are turned over one at a time. Let $B_n$ denote the number of black cards left just before the $n^{th}$ card is turned over. Let \[ Y_n = \frac{B_n}{r+b-(n-1)}\,. \] (So $Y_n$ equals the proportion of black cards left just before the $n^{th}$ card is revealed.) Show that $Y$ is a martingale.
Suppose $N_1, N_2, \dots$ are independent identically distributed normal random variables of mean $0$ and variance $\sigma^2$, and put $S_n=N_1+\ldots+N_n$.
1. Show that $S$ is a martingale.
2. Show that $Y_n= \exp\left(S_n - \tfrac{n}{2}\sigma^2\right)$ is a martingale.
3. How should these expressions be altered if $\operatorname{\mathbb{E}}\left[N_i\right] = \mu\neq 0$?
Let $X$ be a discrete-time Markov chain on a countable state-space $S$ with transition probabilities $p_{x,y}$. Let $f: S \to \mathbb{R}$ be a bounded function. Let $\mathcal{F}_n$ contain all the information about $X_0, X_1, \ldots, X_n$. Show that \[M_n = f(X_n) - f(X_0) - \sum_{i=0}^{n-1} \sum_{y \in S} (f(y) - f(X_i)) p_{X_i,y}\] defines a martingale. (Hint: first note that $\operatorname{\mathbb{E}}\left[f(X_{i+1}) - f(X_i) | X_i\right] = \sum_{y \in S} (f(y) - f(X_i)) p_{X_i,y}$. Using this and the Markov property of $X$, check that $\operatorname{\mathbb{E}}\left[M_{n+1} - M_n | \mathcal{F}_n\right] = 0$.)
Let $Y$ be a discrete-time birth-death process absorbed at zero: \[ p_{k,k+1} = \frac{\lambda}{\lambda+\mu},\quad p_{k,k-1} = \frac{\mu}{\lambda+\mu}\,, \qquad \text{for $k>0$, with $0<\lambda<\mu$.} \]
1. Show that $Y$ is a supermartingale.
2. Let $T=\inf\{n:Y_n=0\}$ (so $T<\infty$ a.s.), and define \[X_n = Y_{\min\{n, T\}} + \left(\frac{\mu-\lambda}{\mu+\lambda}\right)\min\{ n, T\} \,.\] Show that $X$ is a non-negative supermartingale, converging to \[ Z=\left(\frac{\mu-\lambda}{\mu+\lambda}\right)T \,. \]
3. Deduce that \[ \operatorname{\mathbb{E}}\left[T|Y_0 = y\right] \leq \left(\frac{\mu+\lambda}{\mu-\lambda}\right)y\,. \]
Let $L(\theta; X_1, X_2, \ldots, X_n)$ be the likelihood of parameter $\theta$ given a sample of independent and identically distributed random variables, $X_1, X_2, \ldots, X_n$.
1. Check that if the “true” value of $\theta$ is $\theta_0$ then the likelihood ratio \[M_n = \frac{L(\theta_1; X_1, X_2, \ldots, X_n)}{L(\theta_0; X_1, X_2, \ldots, X_n)}\] defines a martingale with $\operatorname{\mathbb{E}}\left[M_n\right] = 1$ for all $n \ge 1$.
2. Using the strong law of large numbers and Jensen’s inequality, show that \[\frac{1}{n} \log M_n \to -c \text{ as $n \to \infty$.}\]
Let $X$ be a simple symmetric random walk absorbed at boundaries $a<b$.
1. Show that \[ f(x)=\frac{x-a}{b-a} \qquad x\in[a,b]\] is a bounded harmonic function.
2. Use the martingale convergence theorem and optional stopping theorem to show that \[f(x) = \operatorname{\mathbb{P}}\left(X \text{ hits }b\text{ before }a|X_0=x\right)\,.\]