2
votes
2answers
48 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
0
votes
1answer
22 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
2
votes
1answer
23 views

What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
1
vote
1answer
51 views

Two notions of total variation norms

I found these two definitions of the total variation norm for probability measures on $(X,\mathcal{F})$: $$ \left \|\mu- \nu \right \|_{TV} = \sup_{\text{$f:X \rightarrow [-1,1]$ measurable}} \left ...
5
votes
1answer
186 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
0
votes
0answers
17 views

Conditional expectation for discrete random variables

Is it correct that for two discrete random variables $X,Y$ we just have $$E(X|Y \in A) = \sum_{x \in ran(X)} xP(X=x|Y \in A)?$$ This should follow from $$E(X|Y \in A) = \sum_{y \in A}E(X|Y = {y}) ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
1
vote
0answers
35 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
3
votes
0answers
29 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
4
votes
1answer
38 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
1
vote
1answer
55 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
1
vote
0answers
39 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
0
votes
1answer
29 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
27 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
0
votes
1answer
39 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
1
vote
1answer
32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
3
votes
0answers
58 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
0
votes
0answers
20 views

Understanding certain parts of the proof of Helly's Selection Theorem

I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ...
1
vote
1answer
27 views

Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?

Suppose that we know that $ \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) $ for every probability measure $\mu \in \mathcal{A}$ in a certain class. Also, suppose that $\{\nu_{n}\}$ ...
1
vote
0answers
23 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
38 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
3
votes
0answers
104 views

Question about Feller's book on the Central Limit Theorem

My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller's book 'An Introduction to Probability Theory and its Applications'. Theorem 1 proves the Central Limit Theorem in the ...
2
votes
0answers
85 views

Kolmogorov extension-type result

I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let $(\Omega, \mathscr{F}, P)$ denote our ...
3
votes
1answer
63 views

Topology of weak convergence

Edited: Thanks to etienne. I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak ...
4
votes
1answer
85 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
0
votes
1answer
22 views

Find a asymptotic upper bound for $\sum_{n=N}^{\infty}p_{ii}^{(n)}$ for a asymetric one-dimensional simple random walk

For asymmetric one-dimensional simple random walk, that is $$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)$$ for some $p \ne 1/2$, provide an asymptotic upper bound for ...
1
vote
1answer
25 views

Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
0
votes
1answer
67 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
1
vote
1answer
78 views

Relation between convergence in distribution and weak convergence

If $(X_n: n\in \mathbb{N}), X$ are a sequence of random variables in $\mathbb{R}$, I wish to show that $X_n \to X$ weakly if and only if $X_n \to X$ in distribution. By 'converging weakly' I mean that ...
0
votes
0answers
54 views

A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
1
vote
1answer
114 views

Expected value, I do not get this “wikipedia triviality”

on wikipedia are two definitions of the expected value for some random variable $E(X)=\int t f_X(t) dt$ and $E(X)=\int X dP$. I do not see how they are equivalent. In cases, that $X$ would be ...
1
vote
2answers
118 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
0
votes
1answer
65 views

Inequalities of the quantile function [closed]

I'm trying to rigorously prove the following inequalities involving the quantile function $Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\}$ where $F$ is the distribution function: 1) $F(x) < a \iff ...
0
votes
0answers
24 views

Problem with an inequality from probability theory (Random matrix theory)

I read the following notes on random matrix theory http://www.umpa.ens-lyon.fr/~aguionne/cours.pdf . While reading Wigner's proof for the semi-cicular law I encoutered the following inequality on page ...
1
vote
1answer
29 views

Convergence of the limit of a sum of $|P_{nk}-P_{k}|$, where $P_{nk}$ and $P_{k}$ are sequences of nonnegative numbers summing to 1

Let $(P_{nk})_{k \geq 1}$, $n=1,2,\cdots$ and $(P_{k})_{k \geq 1}$ be a sequence of nonnegative numbers satisfying $\sum_{k=1}^{\infty}P_{nk}=1$ and $\sum_{k=1}^{\infty}P_{k}=1$, and let $\lim_{n ...
1
vote
1answer
48 views

supremum norm and convergence.

Suppose $f:[0,\infty) \rightarrow \mathbb{R}$ is continuous. Suppose that for some $\epsilon$ > 0, $\max_{t \in [0,n]} |f(t)|$ < $\epsilon$ for all $n \in \mathbb{N}$. Is it then true that ...
1
vote
1answer
47 views

IFF conditions for convergence in probability and almost surely

I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ...
1
vote
1answer
64 views

Law of large numbers for Brownian Motion

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? ...
1
vote
1answer
38 views

Definition of a random variable in the context of a hypergeometric distribution

We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used ...
0
votes
1answer
149 views

Convergence of events in a probability space with respect to $L^2$

Define for events $X, Y$ that $d(X,Y) = P((X-Y) \cup (Y-X)) $ = $ P(X \bigtriangleup Y) $, show that $d(X_n,X) \rightarrow 0$ if and only if $\chi_{X_n}$ converges in $L^2$ to $\chi_X$ (these are ...
3
votes
1answer
56 views

Does convergence of $\lim_{n \rightarrow \infty} \int h d\mu_n$ for all continuous, bounded $h$ imply weak convergence of $(\mu_n)$?

Let $(\mu_n)$ be a sequence of positive finite Borel measures on $\mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} \int h d\mu_n$ converges for every bounded, continuous function $h$ on $\mathbb{R}$. ...
2
votes
1answer
182 views

Does weak convergence with uniformly bounded densities imply absolute continuity of the limit?

Suppose $(f_n)$ is a sequence of probability density functions on $\mathbb{R}$ such that \begin{align*} f_n &\leq M \text{ for all }n \\ f_n(x) &= 0 \text{ for all } |x| > 1 \end{align*} ...
5
votes
1answer
104 views

Proving $\int^{\infty}_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0$

I've been asked to prove that $$ \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$ My approach so far has been to use a theorem proved in class ...
2
votes
1answer
75 views

Techniques for determining whether the weak limit of an absolutely continuous sequence of probability measures is absolutely continuous?

Let $(\mu_n)$ be a sequence of probability measures on $\mathbb{R}$ that converges weakly to a probability measure $\mu$ on $\mathbb{R}$. So $$ \lim \int hd\mu_n = \int h d\mu $$ whenever $h$ is a ...
2
votes
1answer
157 views

Show using dominated convergence that summation and differentiation can be interchanged.

Problem: Define $\{p_k: k \ge 0 \}$ as a probability mass function on a discrete random variable X taking values on $\{0,1,...\}$ then define the generating function $P(x) = E(x^X) = ...
0
votes
0answers
88 views

Probablity - A question about Poisson distribution and Stirlings Formula leading up to the central limit theorem

The goal of this problem is to prove that $P(S_n = k)\sqrt{2\pi n} \rightarrow e^{-\frac{x^2}{2}}$ by using Stirling's formula. Here is what is given: 1) $S_n = \sum_1^n X_i$, where $\{X_i\}$ are ...
5
votes
0answers
103 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
2
votes
0answers
55 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
0
votes
0answers
42 views

Chernoff bound in binomial case

Let $S_n$ the binomial distribution with parametres $n,p$. I have to prove that $$P(S_n\geq n(p+\varepsilon))\leq e^{-2n\varepsilon^2}$$ for every $\varepsilon\geq 0$. I have to use Stirling's ...
0
votes
0answers
56 views

Exercise 3.2.5 Durrett

"Suppose $g, h$ are continuous with $g(x)>0$ and $|h(x)/g(x)|\rightarrow 0$ as $|x| \rightarrow \infty$. If $F_n =>F$ and $\int g(x) dF_n\leq C< \infty$ then $\int h(x)dF_n(x)\rightarrow ...