Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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60 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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100 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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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 ...
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1answer
53 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
43 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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459 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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108 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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16 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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25 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
73 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 ...
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1answer
93 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
24 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. ...
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118 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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86 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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33 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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93 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
43 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
42 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 ...
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161 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 ...
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1answer
118 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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1answer
35 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 ...
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1answer
31 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 ...
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63 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 ...
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1answer
95 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 ...
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1answer
85 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 ...
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10 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 ...
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4answers
1k 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. $$E[\sin\theta \cos\theta] = \ ?$$
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1answer
147 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: ...
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1answer
34 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( ...
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52 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 $x$.
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46 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 ...
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413 views

Problem on EU commission

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 ...
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1answer
93 views

Conditional expectation almost sure

If $X_1 = X_2$ on a measurable set $B \in \mathfrak F$ then $E(X_1\mid\mathfrak F)=E(X_2\mid\mathfrak F)$ almost sure on $B$.
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351 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 ...
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1answer
49 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 ...
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1answer
65 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 ...
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50 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 ...
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72 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 ...
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1answer
39 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 $$ ...
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1answer
75 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, ...
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64 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 ...
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59 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 ${[X_n]}_{n\geq1}$. ...
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1answer
24 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 ...
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1answer
63 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 ...
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2answers
137 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 ...
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33 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 ...
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38 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: $$ ...
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1answer
20 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 ...
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1answer
153 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 $\phi_n$ ...
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2answers
25 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 ...