2
votes
1answer
32 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
43 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
54 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
59 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
82 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
34 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
65 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
32 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
89 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
36 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
38 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 ...
0
votes
1answer
13 views

Advanced probability-limit related to non-negative random variables.

how to show the above two limits equal to 0?Thanks for helping me.
1
vote
1answer
35 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
61 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
78 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
38 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 ...
0
votes
1answer
68 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
132 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
71 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
52 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
68 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
60 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
108 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
66 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
177 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
118 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
99 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
85 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
37 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
230 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 ...
8
votes
1answer
418 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
68 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
146 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
295 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
50 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
844 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 $t$, $\{ ...
1
vote
2answers
143 views

One question regarding r.v independence

I just encounter independence in a Statistics course, I get stuck in this question for a long time..any help will be extremely appreciated. First one is, if $X_1, X_2, X_3...X_k$ (finitely many) are ...
0
votes
1answer
171 views

Interchanging the order of limit and expectation

Assume $\displaystyle\lim_{t\to0}X_t=\gamma\hspace{3pt}a.s.$ where $X_t\geq 0$. I would like to show that $\displaystyle\lim_{t\to0}E[X_t]=E[\lim_{t\to0}X_t]=\gamma$, i.e. that it's possible to ...
1
vote
1answer
162 views

Poisson distribution and probability of random variables

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
0
votes
1answer
70 views

Poisson distribution and probability distributions

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
1
vote
1answer
181 views

How Borel sets make $\sigma$-algebra in a topological space?

I am trying to wrap my head around random variables and can't prove the following questions: How, in a topological space $(X, \mathcal{T})$, the collection of all Borel sets, say $\mathfrak{B}$, ...
1
vote
1answer
698 views

Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
0
votes
2answers
137 views

Can we use Fubini's Theorem?

Is there any special technique to deal with the distribution of sum of two random variables where they are not independent? For example I have concluded that if $X =_p W$ and $Y=_pZ$ ($=_p$ means ...
1
vote
1answer
175 views

What is the sigma algebra generated by the indicator function of random variable?

I was thinking what is the $\sigma$-algebra generated by the random variable $Z= \mathbb{I}(X+Y=0)$ where $X,Y\sim\operatorname{Bern}(p)$ iid. (Note: $\mathbb{I}$ is the indicator function.)
4
votes
1answer
211 views

Combinations of i.i.d Inverse Chi-Square RVs and their characteristic functions

I am working on a few self-study problems in probability/measure theory and am stuck on characteristic functions. I have the following problem: Given: $X_1,\ldots,X_n$ are iid inverse chi-square(1) ...
1
vote
1answer
559 views

Sums of independent random variables converging almost surely

I am working through Achim Klenke's text entitled "Probability Theory", and I came across the following interesting exercise: Let $(X_i)_{i\in\mathbb{N}}$ be independent, square-integrable random ...
0
votes
1answer
124 views

Elementary question on random variables and Borel measurable mappings

In probability theory we have this definition: DEFINITION: Let $(\Omega, \mathcal{U}, P)$ be a probability space. A mapping $\mathbf{X}: \Omega \to \mathbb{R}^n$ is called an n-dimensional random ...
-2
votes
1answer
149 views

Radius of convergence of a series of random variables

Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, ...
2
votes
1answer
323 views

$\sigma$ algebra of collection of random variables

Im doing a course on measure theory and I'm stuck on one of the exercises. Take $\{Y_{\gamma}:\gamma \in C\}$ as an arbitrary collection of random variables and $\{X_{n}: n \in N\}$ to be a countable ...
1
vote
1answer
222 views

Under what conditions is expectation value distributive?

We know that for two real numbers $a,b$ and two random variables $X,Y$ we have that $E(a X + b Y ) = a E(X) + b E(Y)$. Under what conditions is it also true that for any three random variables ...