数学代写|MAST31702 Probability Theory 👍 👍 - 👍代考代写：100%准时可靠您的作业代写专家

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Prerequisites

Prerequisites
Probability theory I and its prerequesites

Learning outcomes
The main point is to study the principal tools in modern Probability, namely, conditional expectations and martingales, together with some examples

Learning materials
Lecture notes; D. Williams: “Probability with martingales”, R. Durrett: “Probability: theory and examples”

Additional info
Target group
Optional course.

Master’s Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Timing
Recommended time/stage of studies for completion: 1. year

Term/teaching period when the course will be offered: varying

Completion methods
Exam, other methods will be described later

MAST31702 Probability Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

(a). Let $X_1, X_2, X_3, \ldots$ be i.i.d. with
$$
P\left(X_1=(-1)^k k\right)=C /\left(k^2 \log k\right)
$$
for each $k \geq 2$ and where $C$ is chosen so the probabilities add to 1 . Show that the expected value of $X_1$ is infinite but nonetheless there exists a finite $\mu$ so that
$$
S_n / n \rightarrow \mu
$$
in probability.
(b). Find a distribution with the property that if $X_1, X_2, X_3, \ldots$ are i.i.d. with this distribution, then $x P\left(\left|X_1\right| \geq x\right.$ ) goes to 0 as $x \rightarrow \infty$ (which implies that there is a type of weak law of large numbers as proved in class) but such that
$$
S_n / n \rightarrow \infty
$$
in probability.

问题 2.

Construct an example of independent random variables, $X_1, X_2, X_3, \ldots$, such that all the distributions are symmetric, all have infinite mean but
$$
\sum_n X_n
$$
converges a.s.

问题 3.

Let $X_1, X_2, X_3, \ldots$ be i.i.d. mean 0 , variance 1 random variables.
a. Show that for each $\mathrm{t}$,
$$
\sum_n X_n \frac{\sin (n t)}{n}
$$
converges a.s.
b. Show that it follows from part (a) that
$$
P\left(\sum_n X_n \frac{\sin (n t)}{n} \text { converges for every rational } t \in[0,1]\right)=1 \text {. }
$$
c. Show that it follows from part (a) that
$$
P\left(\sum_n X_n \frac{\sin (n t)}{n} \text { converges for Lebesgue a.e. } t \in[0,1]\right)=1 \text {. }
$$
Hint: Fubini’s Theorem. You may assume, in the application of Fubini’s Theorem, that the relevant sets are measurable without proving that.
d. Explain why you are not necessarily able to conclude that
$$
P\left(\sum_n X_n \frac{\sin (n t)}{n} \text { converges for every } t \in[0,1]\right)=1 .
$$
The latter turns out to be true nonetheless but it is a nontrivial theorem. The general theme of this question (when are there exceptional values of $t$ for something to occur even though it holds a.s. for each fixed $t$ ) is a question which permeates alot of probability theory and we will see some of this when studying Brownian motion.

问题 4.

Let $X_1, X_2, X_3, \ldots$ be independent and $S_n$ be, as usual, the partial sums. Which of the following two statements are true? Prove or give a counterexample.
(i). If $X_i$ converges to 0 a.s., then $S_n / n$ converges to 0 a.s.
(ii). If $X_i$ converges to 0 in probability, then $S_n / n$ converges to 0 in probability.

问题 5.

Let $g(x)=x^2$ for $|x| \leq 1$ and $=|x|$ for $|x| \geq 1$. Show that if $X_1, X_2, X_3, \ldots$ are independent with $E\left(X_i\right)=0$ for each $i$ and
$$
\sum_n E\left(g\left(X_n\right)\right)<\infty
$$
then
$$
\sum_n X_n
$$
converges a.s.