5
votes
2answers
38 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
0
votes
1answer
33 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 ...
0
votes
0answers
41 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 ...
1
vote
1answer
36 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
0
votes
1answer
15 views

quick question on measurability of random variables and what becoming a deterministic function means.

we stated a theorem in class: if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y. This is fine. The Professor sometimes states that X ...
1
vote
2answers
43 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
7
votes
0answers
72 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
2
votes
1answer
22 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
2
votes
1answer
48 views

Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
1
vote
1answer
30 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
1
vote
1answer
30 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
0
votes
1answer
30 views

$\sigma(Y)$-measurable R.V. $X$ and Borel functions

I have to prove that if $Y: \Omega \rightarrow \mathbb{R}$ then $X: \Omega \rightarrow \mathbb{R}$ is a $\sigma(Y)$-measurable function if and only if exists a Borel function $f: \mathbb{R} ...
6
votes
0answers
182 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 ...
2
votes
0answers
77 views

A measure has no point masses: is it absolutely continuous?

I have a question about measure theory. Let $\mu$ be a measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R})$. Assume that $\mu$ has no point masses - i.e. for every $a \in \mathbb{R}$, $\mu({a})=0$. Can ...
1
vote
1answer
27 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
1answer
40 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
3
votes
1answer
63 views

Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a ...
1
vote
2answers
107 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
2
votes
1answer
76 views

inclusion of $\sigma$-algebra generated by random variables

Consider the following random variables $$X:\Omega\to\mathbb{R}\quad\text{and}\quad Y:\Omega\to \mathbb{R}$$ and $$Z:=XY$$. One may interpret it as follows, i.e. $$Z(\omega) = X(\omega)Y(\omega).$$ ...
3
votes
3answers
145 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
1
vote
1answer
43 views

Show: $X_n\xrightarrow{\mathcal{d}} X$, then $\mathbb{E}\lvert X\rvert\leqslant\liminf_{n\to\infty}\mathbb{E}\lvert X_n\rvert$

Let $X_n, X$ be random variables with $X_n\xrightarrow{d} X$. Show that then $$ \mathbb{E}\lvert X\rvert\leqslant\liminf_n \mathbb{E}\lvert X_n\rvert. $$ So let $X_n\xrightarrow{d} ...
0
votes
2answers
87 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
0
votes
2answers
61 views

Show that F(X) and Y-E[Y|X] uncorrelated

X, Y are random variables on probability space , f is a measurable function, Show that F(X) and Y-E[Y|X] uncorrelated. I have tried the followings : Cov(f(X),Y-E[Y|X]) = E[f(X)(Y-E[Y|X])] ...
1
vote
1answer
112 views

Prove that two random variables are almost surely equal

$X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely (a.s) and $X= E[Y|X]$ a.s. Prove that $X=Y$ a.s. The hint I was given is to evaluate : $$E[X-Y;X>a,Y\leq a] + E[X-Y;X\leq ...
0
votes
1answer
38 views

Is $\mathbb{E}\left[ (Y-X)^{2} \right]=0$ equivalent to $\mathbb{P}(Y=X)=1$?

I have a space definied as follows $\mathcal{L}^{2}=\lbrace X: X-\text{real-valued random variable}, \mathbb{E}(X^{2})<\infty \rbrace$ My textbook later states that we don't differentiate between ...
2
votes
2answers
46 views

Function of a set of r.v.'s measurable w.r.t. the $\sigma$-algebra generated by the r.v.'s

Let $X_1, \dots, X_n$ be a set of random variables defined on a probability space $(\Omega, \mathcal{F}, P)$ and denote by $\mathcal{S} = \sigma(X_1, \dots, X_n)$ the $\sigma$-algebra generated by the ...
1
vote
1answer
47 views

Application of the Dominated Convergence Theorem (probabilistic version).

I am currently working on the following problem and I think I've got the solution more or less, but there is a minor question about the usage of the Dominated Convergence Theorem. Let $f: [0,1] ...
1
vote
1answer
63 views

If $Y=\sum_{n=1}^\infty X_n$ diverges, is $Y$ a random variable?

Let $X_n$ be random variables. By definition, a random variable is a function from the probability space to $\mathbb{R}$. If $Y=\sum_{n=1}^\infty X_n$ diverges, is it correct to call $Y$ a random ...
1
vote
1answer
90 views

Series of independent random variables are independent

In the proof of a theorem my lecturer seemed to have used this fact without first proving it: Let $(X_i)_{i \geq 1}$ be real-valued independent random variables on $(\Omega,\mathscr{F},\mathbb{P})$, ...
0
votes
0answers
55 views

$X_n$ independent real random variables such that $S_n$ converges in probability then converges a.s

Let's assume the following result: Proposition: Let $X_n$ be a sequence of independent random variables and let $S_{m,n}=X_{m+1}+X_{m+2}+...+X_n$ (for $m\in$ {$0,1,...n-1$}). Then $P(\displaystyle ...
1
vote
1answer
99 views

convergence in probability induced by a metric

Let $M$ be the set of all random variables from a fixed probability space to $\mathbb R$ with its borel sets. Let's define a metric on $M$ by $d(X,Y)=E(\frac{|X-Y|}{1+|X-Y|})$ I want to prove that ...
2
votes
1answer
140 views

Showing certain functions are random variables

Assume $\{X_k\}_{k \in \mathbb N}$ are random variables on a probability space. Define induced random walk by $S_0 = 0$ and $S_k = \sum_{i=1}^{k}X_i$. Now let $n = \inf\{p > 0: S_p > 0\}$ be ...
0
votes
0answers
164 views

What is the Dirac mass on measure space?

I am reading the book "Lectures on Stochastic Analysis." But I know seldom about measure space. I meet with a symbol which the author call Dirac mass(in 9.3 of this book). Let E be a measurable space, ...
3
votes
1answer
60 views

A sequence of random variables $(X_n)$ such that $\mathbb E(X_n)\to -\infty$ but $X_n\to +\infty$ a.s.

Let $\xi_{1},\xi_{2},\dots$ be random variables (i.e measurable functions) such that $\mathbb{P}(\xi_{n}=-3^{n})=2^{-n}$ and $\mathbb{P}(\xi_{n}=1)=1-2^{-n}$ Let ...
3
votes
2answers
69 views

Real random variable $X$ such that $ \lim_{n\to \infty}{nP(|X|>n)}=0$ what about $ E|X|<\infty$?

Let $X\colon (\Omega,\mathcal F,\mathbb P)\to (\mathbb R,\mathcal B(\mathbb R))$ be a random variable from a probability space to the real numbers with the Borel sets. I proved that if $\mathbb ...
3
votes
1answer
61 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...
1
vote
2answers
112 views

Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?

Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ ...
0
votes
1answer
71 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
3
votes
2answers
220 views

Expectation of composition of functions with density as R-N derivative

In prior probability courses, I've always seen and used the fact that, for a continuous random variable X and a function $\phi$, $E[\phi(X)]=\int_{ \mathbb{R}}\phi(x) f_X(x)dx,$ but I cannot find a ...
2
votes
1answer
137 views

$\sigma$-algebra generated by a random variable

How to show that $\sigma(\{X^{-1}(U):U \text{ open in }R\}) = \{X^{-1}(B): B \text{ Borel set in } R\}$. I can show $\sigma(\{X^{-1}(U):U \text{ open in }R\}) \subseteq \{X^{-1}(B): B \text{ Borel ...
4
votes
2answers
106 views

Measurability problem of sample distribution function of a contiuum of independent random variable

Let $I = [0,1]$ be the index set of a contiuum of i.i.d random variables. For each $t \in I$, the sample space of $X_t$ is $\Bbb R$ equipped with Borel $\sigma$-algebra and Borel probability measure. ...
0
votes
1answer
100 views

Sequences of i.i.d. subgaussian RVs and uniform integrability

Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)? Intuitively it appears to be so; if we take for example $a_j$ i.i.d. ...
1
vote
1answer
39 views

Measurablilty sequence of random variables

When you have a sequence of random variables $\{X_n\}$ measurable with respect to some filtration $\{\mathcal F_n\}$, which converges to some random variable $X$ almost surely. Then what can we say ...
4
votes
1answer
269 views

Prove that it is a random variable iff it is constant on each partition

Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
10
votes
2answers
495 views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
2
votes
2answers
75 views

A question involving Invariant Set in ergodic theorem

I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, ...
3
votes
2answers
192 views

Random Variables that aren't measurable

I've been reading through a math. stats. book, and I'm a little confused with the concept of measurable random variables. The book states: Let $(E, \mathcal{E})$ and $(F,\mathcal{F})$ be two ...
1
vote
2answers
381 views

Definition of atomic $\sigma$-field.

Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
0
votes
1answer
54 views

Question regarding random variable in product probability space

I am struggling at solving product probability space questions, I am wondering if anyone could me with the following question. Let $x_{i}$ be a random variable at probability space ($X_{i}$, ...
1
vote
1answer
971 views

Maximum/minimum of two random variables is a random variable

Suppose $X,Y$ are random variables. I'm trying to understand why $\max\{X,Y\}$ and $\min\{X,Y\}$ are also random variables. The proof in the book that I'm using states that for each $t$, $\{ ...