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 [tag:probability-...

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A simple betting game

Consider the following betting game: Two players each have 100 cents to bet. If one player bets more than the other then that player gains a point and the other player loses a point. The goal of the ...
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92 views

Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
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41 views

Why the moment-generating function, rather than the characteristic function?

I'm wondering why the moment-generating function is worth discussing (say, in basic probability courses, or in textbooks, rather than research), when the characteristic function appears to completely ...
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29 views

Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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27 views

Skorokhod representation proof understanding

The conditions 1-3 were that $F$ is right continuous, $F$ is increasing and the limit of $F(x)$ as $x$ tends to $\infty$ is $1$ and $0$ if $x$ tends to $-\infty$. I don't follow the idea of this ...
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72 views

If $\mathcal{M}_1(E)$ is compact, then $M \subset \mathcal{M}_1(E)$ is exponentially tight?

Let $E$ be a metric space, $\mathcal B$ the Borel $\sigma$-algebra on $E$ and $\mathcal M_1(E)$ the set of probability measures on $(E,\mathcal B)$. I'm interested in when and why the following ...
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51 views

Conditional Expectation.

Let $X$ be a random variable in $L^2(\Omega, \Sigma, P)$ and $\mathcal G$ a sub-$\sigma$-algebra of $\Sigma$. Prove that $E[(X-E[X\mid\mathcal G])^2] \le E[(X-E[X])^2]$. As conditional expectation ...
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80 views

Chernoff vs Berry-Esseen

Let $X_1,X_2,\ldots,X_n$ be iid random variables with mean $\mu > 0$, variance $\sigma^2$ and $X_i \in [-1,1]$ for all $i$. Let $S_n = \sum_{i=1}^n X_i$. Suppose $l < h < 0$. I am trying to ...
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30 views

Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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36 views

Sufficient conditions for Uniform integrability

Let $a_i >0$ be such that $\sum_i {a_i}^2 < \infty$. Define a sequence of random variables $$M_n = \sum_{i=1}^{n} a_i X_i$$. Now I am interested in knowing under what conditions (on $\phi$ and $...
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50 views

If a class of functions $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class?

Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions ...
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71 views

Example 19.6 in van der Vaart: show that the class of indicator functions is P-Glivenko-Cantelli and P-Donsker

I have some doubts related to example 19.6 in van der Vaart "Asymptotic Statistics" which applies Theorem 19.4 (Glivenko-Cantelli) and Theorem 19.5 (Donsker) to the distribution function. Definitions:...
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44 views

Intuition behind the renewal equation

So. I've acquired the unenviable task of having to learn renewal theory on my own. I'm finding most of it to be pretty intuitive, except for one thing. The intuition behind the renewal equation has ...
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32 views

Convergence of conditional expectation

I have the following questions. For random variables $X$ and $Y$ (suppose for simplicty that the are $\mathbb{R}^n$ valued). Now, we have some abstract way of constructing $P(Y\in B|X=x)$ for some ...
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33 views

What is the rate of convergence in the uniform local limit theorem?

Let $(X_i)_i$ be a sequence of iid random variables, s.t. for some sequences $a_n, b_n$ the normalized sum $$Z_n=\frac{X_1+\dots+X_n}{b_n}-a_n$$ converges weakly to an $\alpha$-stable distributed ...
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63 views

If $X_n\nearrow X$ then $E(X_n)\rightarrow E(X)$?

Let $(X_n)$ be an increasing sequence of real valued integrable rvs on a probability space $(\Omega,\mathcal{F},P)$, such that $(X_n)$ converges ae to some rv $X$. Is it true that $E(X_n)\rightarrow E(...
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107 views

Norris exercise: Showing $P_0[\text{no return to}\ 0]=6/\pi^2$

Consider exercise 1.3.4 of Norris' Markov Chains. The question is as follows: Let $\{X_n\}_{n\geq 0}$ be a Markov Chain with state space $S=\{0,1,2,\dots\}$. Suppose the transition probabilities ...
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31 views

Conditional Independence with infinite r.v.'s

I read this property in "Probability with Martingales" from Williams(page 91-92): Suppose that $X_1 , ...,X_r $ are independent, each $X_k$ with law $\lambda_k$. If $h$ is bounded and $\mathcal{B}^...
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31 views

Prove convergence in measure (i.e., in probability) “distributes” over addition and respects nonnegativity.

Suppose $X_{n}$, $Y_{n}$, and $Z_{n}$ are random variables, with $Z_{n} \geq 0$ a.s. and $X_{n} \xrightarrow{p} X$, $Y_{n} \xrightarrow{p} Y$, and $Z_{n} \xrightarrow{p} Z$. Prove the following ...
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42 views

Good book on quantum probability

Who can suggest me a book on quantum probability? I'm mostly interested in its geometric aspects (complex projective space, Fubini-Study...). I have a background in quantum mechanics and differential ...
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126 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
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128 views

Compound Distribution — Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose mean is distributed Normally. ...
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33 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space $(X_t)_{...
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41 views

Poisson distribution transformation method

Random independent variables $x_1, x_2 \sim \ \ \text{poisson(}\lambda )$ $y=x_1+x_2$ $z=x_1-x_2$ The possible density function $f(y,z)=?$ by using Inverse transformation method. Note that I ...
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46 views

Random process involving CDF and PDF of standard normal.

Im studying old exams and found this one: Let $ \Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy $ and $ \phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2} $ be the ...
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45 views

Necessary and Sufficient Conditions for a CDF

This is an attempt to prove Theorem 1.5.3. in Casella and Berger. Note that the only things that have been proven are really basic set-theory with $\mathbb{P}$ (a probability measure) theorems (e.g., ...
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41 views

Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
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59 views

Are two dot products of a random variable vector independent?

Let $w,v$ be two different vectors in the finite vector space $Z_p^m$ over $Z_p$ where $p$ is prime. Let $u$ be a vector chosen uniformly at random from $Z_p^m$. Are the random variables $u \cdot w$ ...
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35 views

A sequence converging to 0 in probability times a sequence bounded in probability

I'm trying to prove the following from Lehman's "Elements of Large Sample Theory" Lemma 2.3.1: If the sequence $\{Y_n, n=1,2,\ldots\}$ is bounded in probability and if $\{C_n\}$ is a sequence of ...
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31 views

Convergence in distribution, in $L^p$ and convergence of first and second moments

For some application, we have the following three assumptions about a sequence of Random Variables $X_n$ (with values in $\mathbb{R}^+$, $n \geq 1$: There exists a $X \geq 0$ such that a) $X_n \...
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28 views

Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
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49 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that $E(|X|^{r+2})<\...
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90 views

(dis)prove:$\sup_{F \in 2^{(L^1(S,\mathbb{R}))}}\limsup\sup_{f\in F}|\int f dP_n-\int fdP|=\limsup\sup_{f\in L^1(S,\mathbb{R})}|\int fdP_n-\int fdP|$

Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) - f(y)| \leq d(...
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28 views

Probability of each state less than or equal to $2^{-n}$ for each $n$?

This relates to the hint to exercise 3.6.5 in Rosenthal's A First Look at Rigorous Probability Theory. ($\Omega, \mathscr{F}, P$) is a probability triple, where: $\Omega$ is countable (in this book, ...
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67 views

Characterization of renewal processes, and examples with exactly one of stationary or independent increments

A continuous-time process $\{N(t) : t\in[0,\infty)\}$ is a counting process if it satisfies $N(t)\geqslant0$ a.s. (nonnegative) $\mathbb P(N(t)\in\mathbb N\cup\{0\}) = 1$ (integer-valued) If $s\...
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68 views

Suppose $X$ and $Y$ have joint density $f(c,y)=6xy^2$ for $x,y\in(0,1)$. What is $P(X+Y<1)$?

My attempt: $$f(x,y)=6xy^2$$ $$P(X+Y<1)=\int_{0}^1\int_{0}^{1-y} 6xy^2dxdy$$ $$=\frac{1}{10}$$ However, the answer was supposed to be $\frac{3}{5}$, and the bounds on the integral were supposed ...
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20 views

The boundedness of a certain sequence of expectations

In Bálint Tóth's paper, "No More Than Three Favourite Sites for Simple Random Walk", while proving one of the many technical lemmas in his theorem's proof, he makes the following claim: suppose for ...
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48 views

Find $P\left(X ≤ \frac12, Y≥\frac34\right)$. Joint Probability Density Function

Textbook: Mathematical Statistics with Applications Wackerly Let $X$ and $Y$ have the joint probability density function given by $$f(x,y) = \cases{6(1-y), \quad 0≤x≤y≤1 \\ 0, \quad \text{elsewhere}...
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81 views

Decomposition of mutual information for conditionally independent variables

I have a question regarding the mutual information of conditionally independent random variables (observations). Given $p(x,y|z) = p(x|z)p(y|z)$ where $z$ corresponds to a latent variable, I was ...
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44 views

If $X_n \rightarrow^{P} 0$ then for any $p >0$ $\frac{|X_n|^p}{1+|X_n|^p} \rightarrow_{P} 0$

Let $\{X_n\}$ be a sequence of random variables If $X_n \rightarrow^{P} 0$ then for any $p >0$ $$\frac{|X_n|^p}{1+|X_n|^p} \rightarrow_{P} 0$$ and $$E(\frac{|X_n|^p}{1+|X_n|^p}) \rightarrow 0$$ . ...
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26 views

Convergence of generalized inverses

I copy-paste the post from Math Overflow - maybe someone can give me a tip. During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation ...
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58 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = F_{\lambda}(\eta_k)$. ...
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42 views

Concentration inequalities for $P(\sum_{i=1}^n \epsilon_i X_i > t)$

Let $\epsilon_i \sim \text{Bernoulli}(p)$ and $X_i \sim \text{Normal}(0, \sigma^2 / n)$ for $i=1,\ldots,n$. I am interested in getting a sub-Gaussian type upper bound for $$ P\left(\sum_i \epsilon_i ...
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36 views

$S_n \in [-a,a]$ for some $a$ infinitely often

Suppose we have iid r.v.s $X_n \in \mathbb{R}$ with mean $0$ variance $\sigma^2$, I wonder is it true that we have $\exists a>0,$ $$P(S_n \in [-a,a] \text{ infinitely often} )=1,$$ where $S_n =\...
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59 views

Understanding Markov's inequality

Here is a statement of Markov's inequality. Suppose that $X$ is a random variable and that $g: \mathbb{R}\to [0,\infty]$ is Borel measurable and non-decreasing. Then, for any real $c$, $$ E[g(X)]...
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25 views

Conditional expectation and stopping time $\mathbb{E}(X1_{T\leq m}|\mathcal{F}_{T\wedge m})=\mathbb{E}(X1_{T\leq m}|\mathcal{F}_T)$

Let $X$ be a random variable and $T$ a stopping time in a filtrated probability space. If $m > 0$ is it true that: $$\mathbb{E}\left(X1_{T\leq m}|\mathcal{F}_{T\wedge m}\right)=\mathbb{E}\left(X1_{...
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23 views

Application of Laplace transform to stopping times and expectations

Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n \...
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38 views

Why the case of independence of random variables is more important than any other specific type of dependence?

Maybe a stupid question but why is the case of independence of, say, two random variables $X$ and $Y$ is in some ways considered to be more ``central'' or more important than any other type of fixed ...
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39 views

Random variable, diagonalization type argument?

Let $d_j$ be i.i.d. random variables, with$$P(d_j = 0) = P(d_j = 1) = {1\over2}.$$Define$$X = 0.d_1d_2 \dots = \sum_{j = 1}^\infty {{d_j}\over{2^{j}}}.$$I know how to show that $X$ is uniformly ...
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46 views

Convergence in Probability of a sequence whose variance is going to 0

Let $X_n \ge 0$ be a sequence of random variables (which are not necessarily independent) satisfying the following conditions: $E(X_n) = \mu_n$ where $l \le \mu_n \le u$ for some constants $l, u <...