1
vote
0answers
10 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
1
vote
0answers
32 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
1
vote
1answer
34 views

Meaning of $P(Y|X=x)$

Suppose that $X$ and $Y$ are two random variables on $(\Omega, \mathcal H, P)$ with values in $(\mathbb R,\mathcal B_{\mathbb R})$. I want to understand what is "formally" the expression $P(Y|X=x)$ ...
1
vote
1answer
35 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
0
votes
1answer
24 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
votes
1answer
27 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
2
votes
3answers
99 views
+50

Probability of events in an infinite, independent coin-toss space

I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the ...
2
votes
1answer
45 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
0
votes
1answer
20 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
0
votes
2answers
26 views

A proposition about positive random variables and expected values

I have problems to give a proof for the following proposition: Consider a random variable $X$ with values in $[0,+\infty]$. If $P(X=+\infty)>0$, then $E(X)=+\infty$ (notation: $E(X)=\int X ...
4
votes
2answers
142 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
22 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
1
vote
1answer
13 views

Limit distribution is invariant

Consider a homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with a countable (but not necessarily finite) state space $S$. Suppose that there exists a limit distribution $\pi$, namely: ...
2
votes
1answer
49 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
0
votes
0answers
38 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
2
votes
1answer
70 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
1
vote
1answer
105 views

Real valued random variables and cumulative distribution functions (c.d.f.)

Let $X$ be a random variable with values in $\mathbb R$ (we fix the Lebesgue measure on $\mathbb R$), then is well defined a c.d.f. $F_X$ such that $$F_X(x)=X_\ast P(]-\infty,x])=P(X\in]-\infty,x])$$ ...
0
votes
1answer
42 views

An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
0
votes
1answer
29 views

A field being a sigma field if and only if it's a monotone class

The exercise is as follows: "The limit of an increasing (or decreasing) sequence An of sets is defined as its union ∪nAn (or the intersection ∩nAn). A monotone class is defined as a class ...
0
votes
0answers
24 views

Distribution of an upper limit of a sum of random variables

Let $\{X_{n,j},n\geqslant1,j\geqslant1\}$ be independent and identically distributed random variables. Denote $S_{n,k}=\sum_{j=1}^kX_{n,j}$. Let $\{Z_n,n\geqslant1\}$ be a sequence of random variables ...
3
votes
1answer
40 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
36 views

Convergence of running maximum of uniform random variables [closed]

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
40 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
34 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
57 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
28 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 $(X_i)_{i\in\mathbb{N}}$ ...
2
votes
1answer
62 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
36 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
28 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
72 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
38 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
42 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
17 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
16 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
39 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
58 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
29 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
76 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
84 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
45 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
32 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
33 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
27 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
21 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
29 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
152 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 ...