1
vote
1answer
23 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
-1
votes
2answers
27 views

Convergence of running maximum of uniform random variables [on hold]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
1
vote
1answer
36 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
3
votes
1answer
28 views

countably additive function P

This problem comes from exercise 1.3.5(b) of 'A First Look at Rigorous Probability Theory'. It asks to give an example of a countably additive function $P$, defined on all subsets of $[0,1]$, which ...
0
votes
1answer
55 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
1
vote
1answer
25 views

Showing that a set is in terminal $\sigma$-Algebra

I am reading a probability theory book (from Bauer) and I found the following statement in the book that I cant understand: Given a sequence of independent random variables $(X_i)_{i\in\mathbb{N}}$ ...
1
vote
0answers
35 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
0
votes
1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
2
votes
1answer
22 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
0
votes
1answer
71 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
2
votes
1answer
33 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
0
votes
1answer
41 views

Prove that a Modified Cantor Distribution is Atomic.

Consider a measurable space $\{\mathcal{I},\mathcal{B}\}$, where $\mathcal{I} = [0,1]$ and $\mathcal{B}$ are the Borel sets on $\mathcal{I}$. And also, denote $\mathcal{C}$ as the cantor set on ...
0
votes
0answers
19 views

Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
0
votes
1answer
16 views

Vantage point tree question

I'm stuck in understanding the 1993 vantage point tree paper: http://aidblab.cse.iitm.ac.in/cs625/vptree.pdf It defines some things first: So if $x\in[0,1]$, then $P(x)$ is the probability of the ...
0
votes
0answers
14 views

Filtration generated by a specific family of random variables

I need help with this problem: Let $X$ be a uniform random variable on $(0,1)$ i.e for every measurable set $A$ $$\mathbb{P}(X \in A)=\int_{A \cap (0,1)}dx$$ Let $X_n= 2^{-n}\lfloor 2^n X \rfloor$ ...
0
votes
1answer
37 views

Random variables and integrals

Could someone please explain how this holds: $\displaystyle \int_{\mathbb{R^n}} f d\mu = \int_{\Omega}f(Y_n)d\mathbb{P}$ Does it use the following proposition? Furthermore how does ...
2
votes
2answers
57 views

Independence of Random Variables and Distribution Functions

Let $X_1, X_2,\ldots$ be random variables on $(\Omega, \mathcal{A}, \mathbb{P})$. If $\mathbb{P}(X_1 \leq x, X_2 \leq y)=\mathbb{P}(X_1 \leq x)\mathbb{P}(X_2 \leq y)$ for all $x,y \in \mathbb{R}$. ...
0
votes
1answer
21 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
0
votes
1answer
20 views

do we have $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$?

Let X be a random variable. I can't find a rigorous proof to show that $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$
0
votes
1answer
75 views

What is purpose of these paragraph?

The following paragraphs are from my study notes on probability theory. It is a section within the independence discussion. But to me, they seem to appear here out of blue. I do not understand what ...
1
vote
2answers
79 views

What does $\vee$ mean in set theory?

The following proof is from Probability by Davar Khoshnevisan. There is a symbol $\vee$ in the third sentence of the proof. What does this symbol mean, please? There seems no definition about it in ...
1
vote
1answer
44 views

Independent Events or Random Variables

First recall the following definition of independent random variables. Let $(X_t)_{t \in \mathcal T}$ be a set of random variables, where $\mathcal T$ is an arbitrary index set. Then $(X_t)$ is ...
0
votes
1answer
28 views

Integration with respect to conditional measure?

Let $(X_n)$ be a Markov chain. For $i\in S$ my text defines $$N_i:=\sum_{n=0}^\infty \mathbf 1_{\{ X_n=i \}}$$ and then, as a part of a larger proof, claims that $$\mathbb E_i(N_i)=\sum_{n=0}^\infty ...
2
votes
1answer
30 views

One Corollary of the Kolmogorov Zero-One Law

Here is an application of the Kolmogorov Zero-One Law given in my textbook (a probability path by Resnick page 107-108). It states that the random variables $\limsup_n X_n$ and $\liminf_n X_n$ are ...
3
votes
0answers
26 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
0
votes
1answer
20 views

A question about tail $\sigma $-algebras

How do I show formally that the event $\{w\colon\, \lim_{k\rightarrow\infty} X_k(w)$ exists $\}$ is in the tail $\sigma$-algebra of the sequence $X_1, X_2,\ldots$? Intuitively is quite obvious. The ...
1
vote
0answers
33 views

Finding a probability measure

Could someone helpme with this problem? First, consider the transition kernel in $\mathbb{R^2}\times B(\mathbb{R})$ given by $K(x,A)=U_{S^1}(A-x)$. We can than define an other kernel in ...
1
vote
1answer
19 views

Independence: pi-system lemma proof in Probability with Martingales by Williams

In paragraph 4.2 of Probability with Martingales by Williams the following lemma is stated ($(\Omega, \mathcal{F}, P)$ is a probability triple): LEMMA. Suppose that $\mathcal{G}$ and $\mathcal{H}$ ...
0
votes
2answers
101 views

How can one pass from “almost surely” to “surely”?

Several results (e.g in probability theory or using prob. theory) are stated in an almost surely phrasing (meaning the set of outcomes where this is not so has measure zero) How can one pass from ...
0
votes
2answers
142 views

How to find the following integration

Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess ...
0
votes
0answers
45 views

expectation by integral

I am trying to compute the expectation of $\mathbb{E}[XY]$ where $X$ and $Y$ are dependent on a third non-negative random variable $Z$. I can now compute the expectation as follows ...
0
votes
2answers
44 views

Iterating functions of expectations

We all know that $E[E[X]]=E[X]$. I was wondering, does it also hold that $E[g(E[X])]=E[g(X)]$ for "any" function $g$?
1
vote
1answer
44 views

Noob Question : Need help to understand : Probability with Martingales : page 25

I was asked to post the question from here I am trying to learn measure theoretic probability. The book I am trying to learn it from is Probability With Martingales and, I am really not understanding ...
0
votes
3answers
35 views

$\sigma$-algebra and measurable set.

Today I am reading David Williams's Probability with Martingales. In chapter one, He introduce the notion of Measurable space: A pair $(S,\Sigma)$,where $S$ is a set and $\Sigma$ is a ...
0
votes
1answer
44 views

What is the difference between $\sigma$-algebra, $\sigma$-ring, and field of sets?

I don't really understand the difference between these stuff. They look really similar. What is the difference between those? Which one do people use in measure theory, and probability related things ...
0
votes
1answer
53 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
0
votes
0answers
22 views

absolutely continuous and probability

I have the following definitions Suppose we have $n$ observations $X_{0},X_{1},...,X_{n}$ in $\mathbb{R}$ and the parametrized family of $\{ P_{\theta}: \theta \in \Theta \subset \mathbb{R}^p \}$ ...
6
votes
0answers
168 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
1
vote
1answer
43 views

LimSup of Random Variable

I have a seemingly trivial question. Why does $$\forall a\in\mathbb{R},\mathbb{P}(\limsup X_n>a)>0\Rightarrow \mathbb{P}(\limsup X_n=\infty)=1$$ Clearly, we don't have (at least trivially), ...
2
votes
1answer
31 views

Inequality $||f-g|| < \epsilon \Rightarrow |E[f] - E[g]| < \epsilon$

Let $C(X)$ be the space of continuous bounded functions on some metric space $(X,d)$. Can it be shown that if $||f-g||_\infty < \epsilon$ if follows that $| \int f \, \text{d}P - \int g \, ...
1
vote
1answer
25 views

Expectation of the square of the minimum of iid positive random variables

Let $X_1, X_2$ be i.i.d., positive random variables with $E[X_i] < \infty$ (but $E[X_i^2]$ might be $\infty$). $Y := \min \lbrace X_1, X_2 \rbrace$. I want to show that $E[Y^2] < \infty$. The ...
2
votes
0answers
59 views

Maximal inequality for sums of independent random variables [closed]

Assume that $X_1, X_2, \cdots$ are independent real-valued random variables with partial sums $S_{m} = \sum_{k = 1}^{m} X_k$. How do we show that for any $a>0$ we have $$P(\sup_{m \geq 1} |S_{m}| ...
0
votes
1answer
42 views

Showing $\sigma(X)= \{X^{-1}(B) , B \subset \mathscr{B}\}$

If $X:\Omega \to \mathbb{R}^n $ is any function. $\sigma(X)$ is the smallest sigma algebra generated by all the sets $X^{-1}(U)$, $U\subset \mathbb{R}^n$ open. I am confused as to how you show ...
1
vote
0answers
37 views

Not measurable function whose module is measurable

I read through my notes that is trivial to find a not measurable function $f$ whose module $|f|$ is measurable. However I don't know how to provide such an example.
3
votes
1answer
87 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
1
vote
0answers
42 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
0
votes
1answer
46 views

Continuity of a function defined by means of the Lebesgue measure

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function and $\phi(x)=\lambda ( \lbrace{ t: f(t) >x \rbrace} )$. Prove that $\phi$ is right-continuous but not necessarily ...
1
vote
0answers
37 views

Right-continuity of functions associated to measures

I would like to show that it's possible to associate to a measure a monotone increasing right-continuous function s.t.: $\mu(\left(a,b\right])=F(b)-F(a)$. How can I prove that a function like ...
2
votes
2answers
44 views

Convergence of the probability of a R.V.

I'm trying to solve a certain problem but I'm stuck at a point. The problem is: I have a uniform r.v. $U$ on [$-\pi,+\pi$] and I have a sequence of r.vs $X_1,X_2...$ where $X_k=cos(kU)$. Then I have ...
0
votes
1answer
66 views

Wiener measure integration

Let $\mu_{x,y;t}$ be the Wiener measure generated by $\exp[t \Delta](x,y)$. Now I see in my book the following step: $\int dx\int d\mu_{x,x;t}(\omega) \phi(x) = \int dx\int d\mu_{x,x;t}(\omega) ...