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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Expectation product of pairwise uncorrelated variables

Suppose I have three uncorrelated random variables $X, Y$ and $Z$ (discrete or continuous) such that $$\newcommand{\Cov}{\mathrm{Cov}}\Cov(X,Y)=0;\quad \Cov(Y,Z)=0;\quad \Cov(X,Z)=0 \tag{$\ast$}$$ I ...
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1answer
151 views

Proof of an inequality with expectation of random variables

I'm solving some exercises and got stuck. The setting: let $\varepsilon > 0 $ given, and $D(\varepsilon):=\{(x,y) \in \mathbb{R_+}^2\mid |x-y| \ge \varepsilon\mbox{ and }\min(x,y) \le ...
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192 views

Existence of a random variable $Y$ that minimizes $\lVert X-Y\rVert_2$.

In here, stated as a theorem is: Let $\mathcal{G}\subseteq\mathcal{F}$ be $\sigma$-algebras, $X\in L_2$ be a random variable on the probability space. Then, there exists a random variable ...
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56 views

Finding a sequence of real numbers so $F(M_n-a_n)$ converges to a nontrivial cdf.

Let $\left\{X_n\right\}$ be identically, independently exponentially distributed random variables with parameter/mean $1$. Let $M_n$ be the max of the $X_i$'s up to $n$. Find a sequence ...
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514 views

How is the law of a stochastic process defined?

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process. Let $S^T$ be the collection of all functions from $T$ into ...
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Borderline case for Central Limit Theorem [duplicate]

Possible Duplicate: Limit of sums of iid random variables which are not square-integrable Consider a sequence of iid random variables $X_i$ on $\mathbb{R}$ for which $E(|X_i|^\alpha) < ...
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2answers
202 views

Probability cut-off

Is there a probability cut-off point? e.g. if the probability of something keeps halving do you stop at a certain point and say the rest of the possibilities are negligible? What do you do then? ...
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2answers
365 views

Properties about a certain martingale

I asked this question here. Unfortunately there was not a satisfying answer. So I hope here is someone who could help me. I'm solving some exercises and I have a question about this one: Let $(X_i)$ ...
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1answer
661 views

how to show convergence in probability imply convergence a.s. in this case?

Assume that $X_1,\cdots,X_n$ are independent r.v., not necessarily iid, Let $S_n=X_1+\cdots+X_n$, suppose that $S_n$ converges in probability, how can we show that $S_n$ converges a.s.?
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391 views

A problem related to basic martingale theory

In our probability theory class, we are supposed to solve the following problem: Let $X_n$, $n \geq 1 $ be a sequence of independent random variables such that $ \mathbb{E}[X_n] = 0, ...
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2answers
359 views

Problem concerning continuous probability distribution

How do you prove that the real part of the characteristic function of the continuous probability distribution $f(x)$ is a characteristic function, but the imaginary part is not? The second part is ...
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1answer
98 views

Bound on sum of expectations

Say we have a sequence $\{X_i\}$ independent which is mean $0$ and $E|X_i|^p < \infty$ for $p \geq 1$. Is there a bound in the form $E\left|\sum_{i=1}^n X_i\right|^p \leq C\cdot \sum_{i=1}^n E| ...
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1answer
448 views

Exercise 7.7.1 in Grimmett & Stirzaker's 'Probability and Random Processes'

I'm having trouble solving exercise 7.7.1 in Grimmett & Stirzaker's Probability and Random Processes, which reads: Let $X_1,X_2,\ldots$ be random variables such that the partial sums ...
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1answer
604 views

Understanding a proof of Komlós's theorem

I'm reading a book about probability theory and they use a certain theorem, called Komlós's theorem, which states: For a sequence $ (\xi_n) $ of random variables on $ (\Omega,\mathcal{F},P) $ with ...
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0answers
137 views

Proving that a sequence of random variables satisfies Lindeberg's condition

I have a sequence of iid random variables $X_n$ with zero mean and constant variance $\sigma^2_n$. Let $S_n=\sum_{j=1}^{n}{(j-1)X_j}$. In order to prove asymptotic normality I need to prove first ...
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1answer
137 views

Conditions for convergence to symmetric stable distribution?

Under which conditions converges a sum of i.i.d. random variables $$ \frac{1}{a_N} \sum\limits_{n=1}^N X_n $$ to a symmetric stable distribution? Two examples of sufficient conditions are finite ...
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1answer
183 views

Construction of a characteristic function

Is it possible to construct a characteristic function (for a distribution) $\phi(t)$ such that $\phi(t) = t^{-1/4}$ for $16\leq t \leq 20$? I tried the inversion formula, but didn't know how to ...
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1answer
701 views

Law of Large Numbers and Cauchy Distribution

Let $\{X_n\}_{n=1}^\infty$ be a sequence of iid random variables under the Cauchy distribution with location parameter $0$ and scaling parameter $1$ (so that the density function is $f(x) = ...
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1answer
144 views

Confusion about notation: Is $P(X_2 =1|X_1=3)$ equivalent to $P(X_2 = 1,X_1=3)$?

Assuming $X_i$ is a random variable and $P$ is a probability measure, often I read these two notations in books: $P(X_2 =1|X_1=3)$ $P(X_2 = 1,X_1=3)$ Sometimes even $P(X_2 = 1 \wedge X_1=3)$. Are ...
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1answer
842 views

Linear MMSE estimate of MMSE estimator

This question is prompted by a recent discussion about the relationship between conditional expectation and covariance. Suppose that $X$ and $Y$ are zero-mean unit-variance random variables with ...
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1answer
495 views

How to create a 2d Gaussian Distribution from a 1d Gaussian Distribution

Currently I have a vector of values $x [v_1,\ldots,v_n]$. I can plot the 1d Gaussian by plotting each point $\exp(-(v_i - \mu)^2/(2\sigma^2))\sqrt{2\pi\sigma^2}\cdots$ This gives me a nice looking ...
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1answer
424 views

Conditional Expectation a Decreasing Function Implies Covariance is nonpositive

This comes from the book "A Probability Path". I'm just working through the problems trying to get a grasp of conditional expectations. Suppose $X,Y$ are random variables with finite second moments ...
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1answer
109 views

Convergence of a characteristic function

This is the last part of a three part problem on characteristic functions, and it's been driving me crazy over the last few days. Any help would be most appreciated. $X_1,X_2, \ldots, X_n$ are ...
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3answers
397 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
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1answer
366 views

Joint distribution of non homogeneous Poisson event times?

I am trying to calculate the density of $(T_1,T_2)$ where $T_1$ is the time of the first event and $T_2$ is the time of the second event. I have been looking at the Wiki article on Poisson process and ...
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2answers
490 views

Removing all balls of one or more colors from an urn containing multicolored balls

Let's say you have an urn full of balls. Each ball has one or more colors on it. I'm trying to figure out, given that you draw E balls from an urn without replacement, what is the probability that ...
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72 views

Existence of measure factorization

Let $(\Omega,\mathcal F,\mathbb P)$ a probability space, $(X,\mathcal B)$ a measurable space and $m$ a probability measure on $\Omega\times X$ such that its projection on $\Omega $ is equal to ...
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1answer
112 views

Question about Mutual Information

I am learning about mutual information, and am confused about one of the definitions. Mutual information is defined as $ I(X;Y) = H(X) - H(X | Y) $ where, $$ H(X) = \sum_{x} p(x) \log ...
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150 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...
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42 views

Do matrix concentration inequalities hold when operating on other random vectors

I am aware of concentration inequalities for subgaussian matrices $A$ of the form $\mathcal{P}(\|Ax\|^2 \geq (1+\epsilon)\|x\|^2) \leq \exp(-nc(\epsilon))$. Do these inequalities hold even if $x$ is ...
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122 views

Is the limit of a $L^2$-convergent sequence of random variables unique up to a.e.?

Is the limit of a $L^2$-convergent sequence of random variables unique up to a.e.? In other words, if $X$ and $Y$ are both limits, will $X=Y$ a.e.? If yes, is Ito integral, which is defined as $L^2$ ...
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328 views

Does Itō isometry have different versions?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
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1answer
443 views

recursive equation for number of white balls

Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls ...
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2answers
266 views

Coin toss bet and gamblers, that bet everything

I have come into one problem that I don't know the solution of. Let's suppose there is a casino, where you can bet on toin coss, and you get the double if you bet right. Let's also suppose that the ...
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1answer
126 views

Central Limit Theorem is incorrect - where is my mistake?

Say I flip a coin 80 times and I ask for the probability to get over 48 heads. I then flip a coin 800 times and ask for the probability to get over 480 heads. Translating this into Central Limit ...
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“Small sets” in Markov chains

I came across a definition for a "small set" (of the state space) $A \subset \Omega$: there exists a $\delta > 0$ and a measure $\mu$ such that $p^{(k)}(x, \cdot) \geq \delta \mu (\cdot)$ for every ...
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656 views

how to prove formula of Bayes' rule in multi-variable?

I need to prove this Bayes' rule in multi-variable case: In the probability theorem, if $t_{i}=(y_{i},\delta_{i})$, how could I prove that $p(t_{i}|\theta) = p(y_{i},\delta_{i}|\theta) = ...
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1answer
384 views

Relation between integral by parts and Fubini's theorem

In probability, I have seen some examples for which both Fubini's theorem and integration by parts (for Riemann-Stieltjes integrals with cdf as integrator) provide different but correct solutions. For ...
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153 views

Definition of multivariate martingale

I cannot find a proper definition of multivariate martingale. If each component is $1$-dimensional martingale is it enough for a $d$-dimensional process to be a martingale? Thanks.
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144 views

A Stopping Theorem for Right-Continuous Submartingales

Reading through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (problem 3.24, page 20): Suppose that $ \{ X_t, \mathcal{F}_t \ | \ 0 \leq ...
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1k views

Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable. Let $P$ be the probability measure of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP(x). $$ Its median is defined as a number $m \in \mathbb{R}$ ...
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323 views

Proof for Martingales with Bounded Increments

I'm trying to understand a pretty standard proof about the possible events for Martingales with bounded increments. Specifically, assume 1) $X_1, X_2, \ldots$ is a martingale 2) $|X_{n+1} - X_n| ...
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Probabilistic inequality with two random variables

The problem: Let $\xi,\eta$ be two independent integrable r.v. such that $\mathsf P\{\xi> 0\} = 1$ and $\mathsf P\{\eta\geq 0\} = 1$ and $$ a = \mathsf E[\xi-\eta]>0. $$ Check if ...
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129 views

A question about poisson processes

Reading through the book "Brownian Motion & Stochastic Calculus" by Karatzas and Shreve, I found the following exercise (problem 3.9, page 15): Let $ \ N \ $ be a poisson process with intensity ...
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368 views

Is there a discrete-time analogue of Doléans-Dade exponential?

For a continuous martingale $X$, we have the Doléans-Dade exponential: $$\epsilon(X)_t=\exp\left(X_t-\frac{1}{2}[X]_t\right)$$ What is the "correct" analogue, if one exists, for some discrete-time ...
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4answers
926 views

Random variables: How would you explain it to a beginner?

Different types of random variables: (discrete) Binomial, hypergeometric, geometric, Poisson (continuous) Uniform, normal, exponential Random variables are very useful tools when solving simple and ...
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384 views

A question about stochastic processes and stopping times

Working through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (page 6, Problem 2.2): Let $ X $ be a stochastic process and $ T $ a ...
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79 views

A Question about the Boundedness of the Conditional Expectation of a Random Variable

Assume you are given a probability space $ ( \Omega, \mathcal{ F}, P ) $, a bounded random variable $ X $ on $ ( \Omega, \mathcal{ F}, P) $, and a sub-$\sigma$-algebra $ \mathcal{A} $ of $ ...
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119 views

Bayes rule logic. Why we do use this?

I know that: $$P(c|o) = \frac{P(o|c) P(c)}{P(o)}$$ My question is why can't one calculate $P(c|o)$ directly and not use this formula?