Tagged Questions

Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
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0answers
6 views

Representation Theorem for functionals of Continuous Semimartingales

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable. Does it mean that ...
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1answer
20 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
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0answers
15 views

Lack of memory of a geometric distribution, proving a general case.

I have to prove this for a general value so $P(X > j+k | X>j) = P(X > k)$ Using the conditional probability I get that $P(X > j+k | X>j) = \dfrac{P(X > j+k) \wedge P(X > ...
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0answers
11 views

Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
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1answer
8 views

If $X$ is a random variable satisfying $P[|X| < \infty]=1$ there exist a bounded random variable $Y$ that approximates it well

If $X$ is a random variable satisfying $P[|X| < \infty]=1$, then show that for any $\epsilon>0$ there exist a bounded random variable $Y$ such that \begin{align*} P[X \neq Y] < \epsilon ...
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1answer
20 views

On Measurability of random variable

I am studying measurability of r.v. and I am trying to answer the following question. Suppose $X: \Omega \mapsto \mathbb{R}$ has a countable range $\mathcal{R}$. Show $X \in ...
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2answers
22 views

Conditional Probability - chance for an event to happen

I am learning probabilities at the moment and I have come across this problem: A person takes four tests in succession. The probability of his passing the first test is p, that of his passing each ...
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2answers
14 views

Probability CDF question on highest number of marbles pulled out

I'm kinda stuck on this problem. Here goes: An urn contains n marbles, numbered 1, 2, . . . , n. Suppose k < n marbles are drawn from it at random without replacement. Let X denote the highest ...
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1answer
17 views

Speed of convergence in probability

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $E(X_i)=\mu$. Let $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^{n}X_i$. Let $\{A_n\}_{n \in ...
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1answer
24 views

Application of Slutsky's Theorem

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$. Let ...
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0answers
13 views

Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
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1answer
25 views

Is a measure, which is equivalent to a discrete measure, also discrete?

Let $(\Omega,\mathcal F)$ be a measurable space. Define a probability measure by $\mathbb P=\sum_{k=1}^\infty\alpha_k\delta_{\omega_k},$ where $(\omega_k)_{k\in\mathbb N}\subseteq \Omega,$ ...
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3answers
45 views

Probability problem! Combinations with repetition

guys ! Who can explain this problem?I know that I have to use combinations with repetitions,but also I have to extract some cases.Any ideeas ? : In how many ways can Santa Claus distribute 10 ...
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1answer
22 views

Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not true that $E\left(e^X\right)=e^{E(X)}$. However, when one conditions on $X$, is this true? Does it ...
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0answers
22 views

Gaussian random vector with 0 mean [duplicate]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
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2answers
35 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
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1answer
32 views

How could I proof that there can not be equality in Chebyshev's inequality?

For $k>0$. I have gotten the expresion $F(\mu+\sigma k)-F(\mu-\sigma k) = 1-1/k^2$ for all $k>0$. I can not see why this equality is not possible for any continuous RV, what does this mean for ...
2
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1answer
11 views

distribution of $\|P_VX\|^2$ with orthogonal projection $P$ onto $V$

We've had the following question discussed today but without any result: Let $X_1,\dots,X_d$ be random variables, iid and $X_n\sim N(\mu_n,1)$. How can we describe the distribution of $\|P_VX\|^2$ ...
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2answers
32 views

Can 2 different random variables have the same CDF?

I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. ...
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0answers
87 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
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1answer
31 views

The ito integral is gaussian [duplicate]

Let $\Omega, F, P)$ be the classic setting. I saw that if $f$ is a function which satisfies some assumptions then the integral with respect to the brownian motion is Gaussian. Ie $\int_{0}^{t} f_u ...
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1answer
12 views

$X_n$ are independent rv's. $\sup X_n <\infty$ implies there exist $A$ such that $\sum_{n=1}^{\infty} P(X_n>A)<\infty$

$X_n, n\ge 1$ are independent random variables. Suppose $\sup_n X_n<\infty$ almost surely. Show that there exist $A\in \mathbb{R}$, such that $\sum\limits_{n=1}^{\infty} P(X_n>A)<\infty$ for ...
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1answer
29 views

Conditional expectation knowing $X$ and knowing $f(X)$

I am wondering in which cases the following equality is true: for $X$, $Y$ two random variables and $f$, $g$ two functions, $$\mathrm{Var}\left[\mathbb{E} ( f(g(X),Y) | X ) \right] = ...
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1answer
17 views

weak convergence of a pair of random elements

Consider two sequences $(X_n)$ and $(Y_n)$ of random elements in some nice (e.g. Polish) space s.t. $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ ("$\Rightarrow$" denotes weak convergence). Then we know ...
1
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1answer
20 views

Central Limit Theorem for transformed random variables

The Central limit theorem (CTL) is often given similar to the entry in Wikipedia as: Suppose ${X_1, X_2, ...}$ is a sequence of independent and identically distributed random variables with ...
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1answer
29 views

When to use Central Limit Theorem or Cramers Theorem

In for example this paper the authors say The central limit theorem provides an estimate of the probability \begin{align} P\left( \frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} > x \right) ...
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0answers
5 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
2
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1answer
15 views

Martingale property of product of martingale and stochastic process.

$M_t$ is a martingale with respect to $\mathcal{F _t}$ for $t \geq 0$ and $Z$ is a bounded $\mathcal{F_r}$ measurable random variable. $0\leq r < s <\infty$. I want to show that $Z( M_{s\wedge ...
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1answer
11 views

Resource on Pathwise Computations Involving Brownian Motion

Let $B_{t}(\omega)$ be a standard Brownian motion on $(\Omega,\mathcal{F},\mathbb{P})$. I read in a footnote recently that almost surely the quadratic variation ...
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0answers
25 views

Slutsky's Theorem

In Slutsky's Theorem's proof as outlined in the link, we can get the general results that $g(X_n,Y_n)\rightarrow_d g(X,c)$ whenever $g$ is continuous. However, in the Continuous Mapping Theorem, it ...
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1answer
14 views

The converge of expectation value based on almost sure convergence

Here is the question: Let $\xi_n $ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P) $ such that $E \xi_n^2 \le c $ for some constant $c$. Assume that $\xi_n \to \xi ...
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0answers
11 views

What is Law($Z$) under $\mu$ for a random variable $Z$, and distribution $\mu$?

Is it simply the probability measure $A\in\sigma(\mathbb{R})\mapsto\mu(Z^{-1}A)$? (Or correspondingly whatever the range of $Z$ might be.)
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0answers
21 views

Where have I used the assumption that $X\in L_2$?

Let $X\in L_2$ be a random variable and $g$ a positive real function. Let $I$ be an interval and $b>0$, and suppose that $\forall x\in I\ g(x)>b$. I have to show that: $$\operatorname ...
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1answer
65 views

A basic doubt on the quantity $\ln E[e^X]$

I heard that the quantity $\ln E[e^X]$ expresses variance of $X$ other than $E[X]$. But, I can't prove it formally ? any help will be appreciated. i.e. I want to see how $\ln E[e^X] \geq E[X]$ (other ...
2
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1answer
32 views

Is the limit of a sequence of random variables unique?

If $Y$ and $Z$ are two distinct random variables with the same distribution (for example maybe $Y$ is constant equal to $1$ and $Z$ is equal to $1$ almost everywhere), then surely any sequence $X_n$ ...
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1answer
13 views

Generating set for $\sigma(\mathcal{G}, X)$ where $\mathcal{G}$ is sub sigma field and X is a r.v.

I'm trying to prove the following fact. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-field and let $X : (\Omega,\mathcal{F},\mathcal{P}) \rightarrow (S,\mathcal{S})$ be a random variable. ...
2
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2answers
43 views

Convergence Rate of Sample Average Estimator

Let $X_1, X_2,\cdots$ be i.i.d. random variables with $E(X_1) = \mu, Var(X_1) = σ^2> 0$ and let $\bar{X}_n = {X_1 + X_2 + \cdots + X_n \over n}$ be the sample average estimator. Is there a way to ...
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0answers
14 views

Rate of Convergence in CLT for IID random vectors with dependent entries

Okay, so suppose I've got a sequence of IID random vectors $X_{n}$, where the entries of each $X_{n}=(X_{n,1},\dots, X_{n,i})$ are dependent. The motivating example here is the multinomial ...
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0answers
33 views

Intuition behind conditional expectation $\mathbb{E}[X \mid Y]$ [on hold]

I've read this pretty good answer about intuition behind conditional expectation like $\mathbb{E}[X \mid \mathcal{F}]$ where $X$ is a random variable defined on $(\Omega,\mathcal{A},P)$ and ...
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2answers
26 views

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$. So first I was thinking something along the lines of $$P(R_1 = n, R_2 \leq R_1)$$ would be ...
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0answers
30 views

Covariance of two normally distributed random variables

$$X = \frac1N \sum_1^N d_icos\theta_i $$ $$Y = \frac1N \sum_1^N d_isin\theta_i $$ $$d_i \sim LN(m_i, \sigma^2) $$ (LN mean Log normal distribution, $m_i$ can be measured, actually function of ...
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1answer
40 views

Thy conditional expectation hath forsaken me

Consider the excerpt from below from Tao's book on random matrices (pp.64). I can't understand why the three red underlined expressions are equal. Could you please please please help me ?
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1answer
16 views

Distribution of orthogonal projection onto $\{(x,\dots,x\}\subset \mathbb R^d$

Let $D:=\{(x,\dots,x)\mid x\in\mathbb R\}\subset\mathbb R^d$ and $X$ be a random variable and normally distributed $X\sim N(\mu,\sigma^2 E_d) $ with $\sigma\neq 0$, $\mu\in\mathbb R^d$ and $E_d$ the ...
3
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0answers
54 views

Conditional expectation proof

I hope this is the right place to ask my question - my question comes from some reading I'm doing in mathematical finance, but my question is really a question in probability theory, and is about how ...
4
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0answers
57 views

Converge to Brownian Motion problem

Consider the following sequence of SDEs: $dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$ Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion. I have been ...
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0answers
24 views

Stochastic processes

Update I am a bit confused whether $y_t$ is independent over time under the following assumptions: Consider, first a RV $A$, that follows this process: $A_t = \rho A_{t-1} + e_t$, where $e_t$ is ...
0
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1answer
19 views

How does the following example prove that this set of axioms for a probability field is consistent?

This is froms Kolmogorovs Foundations of probability theory. He gives the following five axioms. Let $E $ be a set and $\mathcal F $ be a set of subsets of $E $. I $\mathcal F $ is closed under ...
2
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1answer
26 views

${P(|X|\geq \lambda)\leq f(\lambda)}$

I have to prove a bound of the form $$P(|X|\geq \lambda)\leq f(\lambda)\quad (1),$$where $f$ denotes some upper bound function and $X$ is a complex random variable. My question is: I know a bound on ...