Arjun Krishnan

MTH 504 Homework

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Homework 6

  1. Le Gall 12.11. This is Rademacher’s theorem

  2. Le Gall 12.14. Another proof of the strong law

  3. Le Gall 12.15 Law of the Iterated Logarithm

Homework 5

  1. Le Gall 12.12. Consider a sequence of with the special property

    Show that it is a martingale, and compute its limit

  2. Le Gall 12.13. Polya’s Urn redux.

  3. Le Gall 12.16. Kakutani’s theorem about products of positive random variables.

  4. Le Gall 12.17. Let be a sequence of independent Bernoulli random variables with . Let be any sequence of positive numbers. Let

    Prove that converges if is in .

    solution

Homework 4

We will continue working through Le Gall’s exercises, all of which are tastefully chosen. We have also found our first typo in Le Gall in Homework 3!

  1. (12.7 Le Gall). Let be a Martingale with , assume that it has bounded increments. Show that almost surely, either it has a finite limit, or if not, both and (it fluctuates wildly).
  2. (12.8 Wald’s identity). Let $X_n$ be iid, and suppose $S_n$ is their usual sum. Suppose $T$ is a stopping time (with finite expectation) so that $S$ is a sum with a random number of elements (this is frequently encountered). Prove that $E[S_T] = E[T] E[X_1]$; i.e., the expected value value of the sum is the expected number of elements times the expected value of each element.
  3. (12.9). Yet another proof of the Strong law. This might be a nice problem for somebody to do.
  4. (12.10). Let $\tau_\infty$ be the tail $\sigma$-algebra of an iid sequence $X_n$. Give a Martingale proof of the Kolmogorov $0$-$1$ law.

Homework 3

  1. Polya’s Urn. An urn contains red and green balls. At each turn, we take a ball out uniformly at random. Then, we replace it and add more balls of the same color. Let be the fraction of green balls.

    1. Show that such that almost surely.
    2. Find the distribution of in the special case where .
    3. Simulate Polya’s urn on a computer and graph

  2. Prove that Doob’s maximal inequalities imply Kolmogorov’s inequalities.

  3. Le Gall 12.2 Prove that and so on.

  4. Le Gall 12.3 Suppose . Then and .

  5. Le Gall 12.4 Let be a simple random walk and satisfy

    Show that is a martingale (among other things).

  6. Le Gall 12.5 Prove that an adapted process is a martingale iff for every bounded stopping time .

  7. Le Gall 12.6 A preview of uniform integrability. Let be a martingale, and let be a stopping time such that

    Prove that . Conclude that .

Homework 2

  1. A (discrete) semimartingale is any process that can be written as where is a martingale, and is a bounded variation process; i.e., $Z_n = U_n - V_n$ where $U_n,V_n$ are adapted processes that are increasing in $n$. Prove that any semimartingale can be written as a difference of a submartingale and a supermartingale.

  2. Suppose are stopping times. Show that

    a.

    b.

    are stopping times.

  3. Let be a stopping time. Let be a martingale. Prove that is integrable and

    in each of the following situations.

    a. $T$ is bounded almost surely (proved in class)

    b. $X$ is bounded and $T$ is $\almostsurely$ finite

    c. and for some ,

  4. Let $X$ be a supermartingale and $T$ be stopping time. Let $C^T$ be the process defined by

    a. Show that $C^T_n$ is previsible. Let

    b. Show that $Y = X_{T \land n}$ and that it is a supermartingale. Use this to show that

  5. Let $X$ be a submartingale, and for $\lambda \in \R$, let

    a. Show that $T$ is a stopping time.

    b. Show that

Homework 1

  1. Prove the following basic properites of conditional expectation. We are working on a space .

    • Tower property: where
    • If is independent of then
    • If is measurable, then almost surely.
    • Let be a sequence of (sub) -algebras and be a sequence of nonnegative measurable random variables. Prove that if converges in probability to , then in probability. Prove that the converse is false.

  2. Let be a collection of iid random variables. If is a tail-measurable function, show that is a constant.

  3. a. If is a random variable with continuous distribution and is such that , show that

    b. (From textbook) Let be an absolutely continuous random varaible with piecewise-continuous density . Prove that for every non-negative measurable ,

    Even though for all , use the preceding to justify the classical definition,

  4. a. If is a martingale and is convex, then is a submartingale if for all . b. If is a submartingale, then is a submartingale if is nondecreasing in addition to convex and integrable.

  5. Suppose is a submartingale bounded in . Show that