# Tagged Questions

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

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
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### 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 ...
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### 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, ...
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### 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. $$Z(\omega) = X(\omega)Y(\omega).$$ ...
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### 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 ...
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### 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])] ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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. ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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 ($X_{i}$, ...
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 $t$, \$\{ ...