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-...

learn more… | top users | synonyms (1)

3
votes
0answers
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}^...
3
votes
0answers
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 ...
3
votes
0answers
39 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 ...
3
votes
0answers
111 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 ...
3
votes
0answers
103 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. ...
3
votes
0answers
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)_{...
3
votes
0answers
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 ...
3
votes
0answers
45 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 ...
3
votes
0answers
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., ...
3
votes
0answers
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: ...
3
votes
0answers
40 views

Clarification on the Augmented Filtration

Consider the following definition. Definition. Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid s\...
3
votes
0answers
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 $u \cdot w$ ...
3
votes
0answers
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 ...
3
votes
0answers
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 \...
3
votes
0answers
27 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 ...
3
votes
0answers
47 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})<\...
3
votes
0answers
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, ...
3
votes
0answers
66 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\...
3
votes
0answers
65 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 ...
3
votes
0answers
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 ...
3
votes
0answers
47 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}...
3
votes
0answers
74 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 ...
3
votes
0answers
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 $$E(\frac{|X_n|^p}{1+|X_n|^p}) \rightarrow 0$$ . ...
3
votes
0answers
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 ...
3
votes
0answers
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 $\xi_k = F_{\lambda}(\eta_k)$. ...
3
votes
0answers
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 ...
3
votes
0answers
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 =\...
3
votes
0answers
57 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)]...
3
votes
0answers
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_{...
3
votes
0answers
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 \...
3
votes
0answers
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 ...
3
votes
0answers
38 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 ...
3
votes
0answers
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 <...
3
votes
0answers
96 views

Probability of combinations of independent events

Disclaimer: I'm almost 70 and trying to learn math concepts I didn't learn earlier in life. So don't expect this to be a deep or particularly difficult question. I've always understood the statement ...
3
votes
0answers
52 views

Repetition in digits on a normal number

Consider a normal real number $0< n < 1$. Let $n_i$ denote the $i^\text{th}$ digit of $n$. What is the probability that there exists some $k \in \mathbb{N}$ such that $n_1=n_{1+k}, n_2=n_{2+k}, \...
3
votes
0answers
94 views

Show the probability of liminfA is 1

Consider a random set $S$ constructed as follows. For any $n>2$, $\mathbb p(n\in S) = 1/\log n$, all independently. Show that $S$ almost surely satisfies the twin prime property: there ...
3
votes
0answers
49 views

Is the term $E\left[Z^3E[Z\mid Y] \right]$ positive or negative?

Let $Z$ be standard normal and $Y$ some other random variable. Is the $E\left[Z^3E[Z\mid Y] \right]$ positive or negative? I tried a few things like using orthogonality principle \begin{align} E\left[...
3
votes
0answers
37 views

Conditional expectation and Joint distribution

Given $(X,Y)$ and $(X',Y')$ pairs of rvs on different probability spaces $\Omega_{1},\Omega_{2}$ respectively but with equal joint distribution show that $E[X|\sigma(Y)]\stackrel{d}{=}E[X'|\sigma(Y')]$...
3
votes
0answers
38 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
3
votes
0answers
224 views

The mathematics of a drinking game called Spoon

This question is about the drinking game Spoon. It was asked on reddit.com/r/math : http://www.reddit.com/r/math/comments/3i9790/drinking_game_turned_mathematical/ The question is whether the person ...
3
votes
0answers
93 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
3
votes
0answers
53 views

Intuition behind almost sure limit of $\frac{|S_n|}{n^{1/p}}$

Suppose $X_i$'s are non-degenerate i.i.d. Then (1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$ (2) However for $p=2$ ...
3
votes
0answers
57 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... $M_\...
3
votes
0answers
54 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in C)...
3
votes
0answers
45 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
3
votes
0answers
89 views

Conditional Distribution absolutely continuous w.r.t Lebesgue measure?

Let $X,Y$ be integrable random variables. Then the condition expectation of $Y$ given $X = x$ is defined as $$ \mathrm E[Y \mid X=x] := \int_\Omega Y(\omega) \, P^X(\mathrm dw \mid x), $$ where $P^X(\...
3
votes
0answers
77 views

Generalizing the pull-out property in conditional expectations

Let $(\Omega, \mathscr{F}, P)$ be a probability space and $\mathscr{G}$ a sub-$\sigma$-algebra of $\mathscr{F}$. If $X$ and $Y$ are (integrable) random variables with $X$ being $\mathscr{G}$-...
3
votes
0answers
61 views

A Simple Inquality with Expected Value

Given $X(k) \ge -1, \;\;k=1,2,3...,N$ be discrete random variable with distribution $f_X$ and assume $a \in [0,1]$ so that $aX(k) \ge -1$ for all $k$ and $$ 1 - E \left[ \prod_{i=1}^N(1+aX(i)) \...
3
votes
0answers
46 views

Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq P(O)...
3
votes
0answers
31 views

Expected value of sorted subsequence

Consider the following discrete random variable: Given an array of size n containing random unique integers, what is the maximal length of a sorted subsequence from that array. What is the expected ...