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 ...

learn more… | top users | synonyms (1)

2
votes
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 ...
0
votes
1answer
66 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
votes
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 $X_n$ ...
1
vote
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
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ?
1
vote
1answer
17 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
votes
0answers
56 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
votes
0answers
61 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 ...
0
votes
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
votes
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
votes
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 ...
1
vote
0answers
22 views

Relative Entropy and variation formula for $C_c$

Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$. By the varational representation formula of Donsker and Varadhan we know ...
0
votes
0answers
22 views

Probability of rolling the same number n or more times in m rolls of a k-sided dice

So the only approach I can find to solve this problem is making computer simulations, anyone can explain a mathematical way to solve it? or recommend a book that can explain this topic. thanks.
1
vote
1answer
32 views

Probability Theory: Conditional Independence and Independence

I have the following definition of conditional independence: $X$ and $Y$ are called conditionally independent given a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$ if for all bounded Borel ...
2
votes
1answer
47 views

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
1
vote
0answers
8 views

Time changes conditions to be adapted

Given a process $X_t$ and another process $T_t$ which is increasing, what conditions should we require such that the process $X_t$ is adapted to the time change $T_t$, that is such that $X_t$ is ...
0
votes
4answers
46 views

Expectation of product of cosine and sine

$\theta\sim U(-\pi,\pi)$. When $\theta$ follows uniform distribution, what is the expected value of the producot of cosine and sine, i.e. $$E[\sin\theta \cos\theta] = \ ?$$
1
vote
1answer
37 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
0
votes
1answer
29 views

How to prove for an arbitrary distributions?

Assume we have some distribution $P(x, y)$ on $\mathbb{X} \times \mathbb{Y}$, how to show that: $$ \mathbb{E}_{x, y} \Bigl[ \bigl( y - \mathbb{E}[y \mid x] \bigr) \bigl( ...
0
votes
3answers
45 views

$\int xe^{-c|x|}\,dx$ where c is a constant

I tried doing integration by parts and saying that $u = x$ and $dv = e^{-c|x|}\,dx$ but then finding $v$ is tricky because of the absolute value of $x$.
1
vote
2answers
32 views

Random variables with the same distribution which difference is positive almost surery.

Let $X$, $Y$ two random variables such that $P(X \le a)=P(Y\le a) \quad\forall a \in \mathbb{R}$ (in other words, $X$ and $Y$ have the same distribution). Suppose $X \le Y$ almost surely. Can I ...
4
votes
0answers
86 views
+200

Voting weights problem

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
2
votes
1answer
26 views

Conditional expectation almost sure

If $X_1 = X_2$ on a measureable set $B \in \mathfrak F$ then $E(X_1\mid\mathfrak F)=E(X_2\mid\mathfrak F)$ almost sure on $B$.
1
vote
1answer
23 views

Prove: Almost sure convergence of random variables with Borel Cantelli lemma

Let $X_n$ be a sequence of random variables with $X_n<\infty$ almost sure for all $n\in \mathbb N$. Show that there are constants $c_n\rightarrow \infty$ such that $\frac{X_n}{c_n}\rightarrow 0$ I ...
0
votes
0answers
29 views

Bounding restriction of random variables

let us restrict a (possibly unbounded) random variable $X$ to those events, such that $X\in(2^{n},2^{n+1}]$ and denote this restricted variable with $\tilde{X}$, i.e. $\tilde{X}:=X\cdot ...
1
vote
1answer
42 views

$\mathbb{E}(e^{tS_{n}})\leq e^{Ct^{2}\sigma^{2}}\quad ?$

Let $S_n=X_1 + \cdots+ X_n$ be a sum of independent random variables such that each $X_i$ has mean zero, variance $\sigma_i ^2$ and lies in $[-1,1]$. Denote with $S_n$ the sum of these random ...
0
votes
0answers
35 views

Weak convergence and convergence in distribution

Is convergence in distribution related to weak convergence in Banach theory? Where by weak convergence I mean: for every functional f the sequence $\langle f,x_n\rangle \overset{n}{\rightarrow} ...
0
votes
2answers
33 views

Conditional expectation and probability

Suppose that we have two real valued Random variables $X,Y$ on a probability setting $(\Omega, F, P)$. Suppose that $X,Y$ have densities $f_X, f_Y$ and joint density $f_{X,Y}$. So I have the following ...
1
vote
0answers
29 views

Measurability properties of processes that arise as limits of sequences of measurable processes

I try to reduce my problem to a more general statement from which I want to know whether this is true in general. I have a sequence of continuous-time stochastic processes $X_t^{(n)}, t \geq 0$ with ...
0
votes
1answer
25 views

Nonhomogeneous Poisson process

Let $\lambda:[0,\infty)\rightarrow [0,\infty)$ be a continuous function and $N$ be a Poisson process with rate $1$. Define $\Lambda(t)=\int_0^t{\lambda(x)dx}$ then how do we prove that $$ ...
0
votes
1answer
53 views

Generalize result on independent RV to conditional independence

Here Independence and conditional expectation is stated that $E(f(X)g(Y))=E(f(X))E(g(Y))$ iff $E(h(X)|Y) = E(h(X))$. Now I'm wondering if this generalizes to independence in the conditional sense, ...
2
votes
2answers
20 views

Completeness of the space of random variables with bounded conditional first moment with respect t0 $\left\Vert \cdot\right\Vert _{2} $

Consider a probability space $\left(\Omega,\mathcal{F},P\right) $, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F} $. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right) $ be the space ...
1
vote
1answer
23 views

Convergence in distribution of the Sum Y/\sqrt{\lambda}

The question follows: ${[X_n]}_{n\geq1}$ is a sequence of independent rv's such that: $P(X_n=-1)=1/2$ $P(X_n=1)=1/2$ Let $N \in Po(\lambda)$ where N is independent of ${[X_n]}_{n\geq1}$. ...
1
vote
1answer
20 views

Determining a Mass Function to $P(X>k+1|X>k) = (k+1)/(k+2)$

I have been struggling with this exercise for a while now and I could a push into the right direction. The exercise is the following: Let $X$ be a random variable which may assume only positive ...
3
votes
1answer
37 views

Proof about independent random variables

Let $X_1,X_2,...$ be independent random variables with $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$ Show that $X_n\rightarrow 0$ in probability if and only if $p_n\rightarrow0$, $X_n\rightarrow 0$ almost ...
4
votes
2answers
69 views

Uniform distribution, as a sum of biased Bernoulli trials.

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
0
votes
0answers
26 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
1
vote
2answers
21 views

$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ ...
0
votes
1answer
17 views

Bounding $\mathbb{E}(X_{i_1}\cdot \ldots \cdot X_{i_k}) $

Consider random variables $X_1,\ldots X_n$ with zero mean, variance at most $1$, $k$-wise independent $k\leq n $ and bounded: $|X_i|\leq C$ for some $C\geq 1$. If I assume $k$ is even, how can I ...
1
vote
1answer
30 views

uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
0
votes
2answers
16 views

Is multiplication normally/binomially distributed?

I was thinking about the binomial formula in the context of coin flips and got to thinking about the reason that even though HHHHHHHHHH is just as likely to occur as a sequence as HHHHHTTTTT, 5 heads ...
0
votes
1answer
16 views

verifying whether a conditional density function is valid

I want to verify whether a given conditional probability function is valid or not. $\mathsf P(y\mid x)=\begin{cases}c\, e^{(-y/x)} & : y\geqslant 0, x>0;\\ 0 & ...
0
votes
1answer
19 views

Introduction to Measure-theoretic Probability George Roussas. example 4 page 1 [closed]

I am reading Introduction to measure-theoretic Probability George Roussas. example 4 page 1 says: Let $\Omega$ be infinite (countably or not) and let $\mathcal{C}= \lbrace A \subseteq \Omega;A$ is ...
0
votes
1answer
27 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let ...
1
vote
1answer
48 views

How can I prove that without further assumptions Chebyshev's Inequality can not be improved?

I have found some examples on the web for specific random variables such $X$ (a discrete type) with probabilities $1/8$, $3/4$ and $1/8$ at the points $x=-1,0,1$ with $\mu=0$ and $\sigma=1/2$. Then, ...