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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4
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2answers
804 views

Dependence and second Borel-Cantelli lemma.

I'll put the problem and then I'll explain my problem. Problem: Let ${A_n}$ be events such as $\operatorname{Cov}(I_{A_i},I_{A_j})=E[I_{A_i}I_{A_j}]-E[I_{A_i}]E[I_{A_j}]\leq 0,\ \forall i\neq j\tag{...
4
votes
1answer
344 views

What's the intuition behind and some illustrative applications of probability kernels?

Given measure spaces $(X, \mathcal{X})$ and $(Y, \mathcal{Y})$ we define measure kernel $\pi : \mathcal{X} \times Y \to [0,\infty]$ such that $\pi(\cdot|y)$ is a measure on $\mathcal{X}$ for every $y \...
4
votes
1answer
94 views

Question on part 3 of the Star Trek problem in Williams, Probability with Martingales

Consider this M.SE question, which is E12.3 in Williams. The answer of Robert Israel (and Xoff) seems to give an exponential bound on $R_n$ almost surely. Wouldn't this imply the convergence of $$\...
4
votes
1answer
132 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let $\Omega=\{1,2,3\}...
4
votes
1answer
609 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 ...
4
votes
1answer
49 views

Stopping time in Markov chains

A random variable $T : \Omega \rightarrow ${$1,2,3...$} $\cup$ {$ \infty$} is called a stopping time if the event {$T=n$} depends only on $X_0 , X_1 ,X_2 ,..., X_n$ for $n = 0,1,2,...$ I have trouble ...
4
votes
2answers
177 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant $...
4
votes
1answer
136 views

Is it that trivial to see that a sequence of random variables is mutually independent?

Grinstead and Snells book, Introduction to Probability, page 144: Here is a number of short questions I have about this text: 0) The authors say that they consider "special classes of random ...
4
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0answers
85 views

The limit in law of a sequence of normal distributions is normal [duplicate]

Let $ \{ \xi_n \}_{n=1}^{\infty}$ be a sequence of normal random variables, where $ \xi_n\sim\mathcal{N}(\alpha_n, \sigma_n^2)$ and $\xi_n \overset{d}{\rightarrow} \xi$. I need to prove, that $\xi$ is ...
4
votes
1answer
1k views

What is the relationship of $\mathcal{L}_1$ (total variation) distance to hypothesis testing?

Kullback-Leibler divergence (a.k.a. relative entropy) has a nice property in hypothesis testing: given some observed measurement $m\in \mathcal{Q}$, and two probability distributions $P_0$ and $P_1$ ...
4
votes
1answer
505 views

Prove that it is a random variable iff it is constant on each partition

Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
4
votes
0answers
875 views

Notation for the pushforward measure

Given a measure space $(X,\Sigma,\mu)$, a measurable space $(Y,\Xi)$ and a measurable map $f\in\Sigma/\Xi$, the pushforward measure $\nu:=\mu\circ f^{-1}$ is given by $$ \nu[A] = \mu[f^{-1}(A)] $$ ...
4
votes
2answers
1k views

Covariance of two random variables with monotone transformation

Suppose I know that, for two random variables $X,Y$, we have $$Cov(X,Y)\neq 0.$$ What happens if we take a monotone transformation of $X$; will the inequality persist? That is, say $f(.)$ is a ...
4
votes
1answer
1k views

Doubts on Mutually exclusive and Independent events

Problem: In a school competition,the probability of hitting the target bu Dick is $\frac{1}{2}$,by Betty is $\frac{1}{3}$ and by Joe is $\frac{3}{5}$.If all of them fire independently,calculate the ...
4
votes
4answers
672 views

The Expectation and the Variance of the runs

folks! I have the following problem: Suppose you have a coin that has chance p of landing heads. Suppose you flip the coin N times and let X denote the number of "head runs" in N flips. A "head run" ...
4
votes
2answers
595 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
4
votes
2answers
562 views

Convergence of random variables (Durrett: Probability Theory and Examples)

I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim): If $X_n$ is ANY sequence ...
4
votes
1answer
172 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
4
votes
1answer
155 views

Expectation conditional on indicator function

Let T and K be dependent continuous random variables, and note the Indicator function as I{.}: Is it correct to say that $E[T|I\{T>t,K<k\}]=E[T|T>t,K<k]$? Is that a property of the ...
4
votes
1answer
148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
4
votes
2answers
624 views

Product of two probability kernel is a probability kernel?

Let $ (\mathbb{X} _i, \mathscr{X}_i) $ and $ (\mathbb {Y} _i, \mathscr {Y} _i) $ measurable spaces with $ i = 1, 2 $. Let $ \gamma_i: \mathscr {X}_i\times\mathbb {Y}_i\longrightarrow [0,1] $ a ...
4
votes
1answer
644 views

Uniform Integrability

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that $$ \displaystyle \...
4
votes
2answers
233 views

Sample path of Brownian Motion within epsilon distance of continuous function

Given a continuous function $f:[0,1]\rightarrow\mathbb{R}$, $f(0)=0$, how can one show that $P(\underset{0\leq t\leq1}{\sup}\left|B_{t}-f(t)\right|<\varepsilon)>0$, where $P$ is the probability ...
4
votes
1answer
61 views

Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$? If the result is not true for ...
4
votes
1answer
174 views

Measurable structure on the space of probability measures

My advisor only half-jokingly mentioned that sometimes people like to consider the measurable structure on $P(X)$ where X is a locally compact polish space and $P(.)$ denotes the probability measures ...
4
votes
2answers
308 views

Do probability measures have to be the same if they agree on a generator of Borel $\sigma$–algebra $\mathcal{B}(\mathbb{R})$?

Suppose $\mathcal{K}\subset 2^\mathbb{R}$ is such that $\sigma(\mathcal{K})=\mathcal{B}(\mathbb{R})$ and let $\mu$ and $\nu$ be measures which agree on $\mathcal{K}$, i.e. $$\mu(A)=\nu(A)$$ for all $A\...
4
votes
1answer
100 views

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
4
votes
2answers
846 views

Simple proof that stationary birth-death chains are reversible

A Markov chain with state space $\mathbb{Z}$ is a birth-death chain if the transition probabilities satisfy $p(x,y) = 0$ for $|x-y| > 1$. That is, the only possible transitions are to move one ...
4
votes
1answer
170 views

A sequence of random variables which converges in distributon converges “to” some random variable

Let $(X_n)$ be a sequence of random variables on a probability space $\Omega$, with distribution functions $F_n$. Suppose $F_n \rightarrow F$ in distribution for some distribution function $F$. Must ...
4
votes
3answers
717 views

Optimally combining samples to estimate averages

Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-...
4
votes
3answers
1k views

Top 3 of 4 Dice Rolls

I'm trying to prove why the mean of the distribution of sums of the top 3 out of 4 fair 6 sided dice is rolls 12.25. Anybody who's rolled a D&D character knows the idea. $r_n = Rand([1,6])$ $x =...
4
votes
2answers
255 views

Dividing by an almost-surely positive random variable

I am reading Shreve's "Stochastic Calculus for Finance II". In it, he states (Theorem 1.6.1) that if $Z$ is an almost-surely strictly positive random variable on a probability space $(\Omega, \...
4
votes
1answer
39 views

A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
4
votes
2answers
87 views

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ ...
4
votes
1answer
309 views

Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and $\mathbb{Q}...
4
votes
1answer
522 views

Counterexample to Jensen's inequality

This appeared in an exam I took. The question asked us to give an example of a convex function $g: \mathbb{R} \longmapsto \mathbb{R}$ and a measure $\mu$ on $\left(\mathbb{R}, \mathscr{B}(\mathbb{R})\...
4
votes
4answers
161 views

Sample: don't confuse measurements with actual values?

In Wikipedia's article on Sample there is the following remark: ''Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these ...
4
votes
1answer
119 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since $...
4
votes
1answer
787 views

A sequence of random variables that does not converge in probability.

I was doing a problem about the converge of the sum of random variables which has two parts: Let $X_1, X_2 ,\dots$ be independent and identically distributed random variables with $E X_i = 0$, $ 0 ...
3
votes
0answers
27 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
1answer
73 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
3
votes
1answer
75 views

Product of random variables convergence in probability

I want to prove that if $X_n\overset{P}\to X$ and $Y_n\overset{P}\to Y$, then $$X_n+Y_n\overset{P}\to X+Y,$$ $$X_nY_n\overset{P}\to XY$$ Proof for the first one I simply use triangle inequality $$ \...
3
votes
3answers
103 views

If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists

Is the following claim correct: If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists, because $$ \mathrm{E} X \leq \mathrm{E} |X| \leq \sqrt{\mathrm{E} |X|^2} $$ by Jensen's ...
3
votes
1answer
171 views

$\lim\inf$ and $\lim\sup$ of events and random variables

I am confused by the $\limsup / \liminf$ of events and random variables. Suppose we have random variables $X_1, X_2, ...$ 1). If $P(X_n >a \text{ i.o.})=1$, we have $P(\limsup\{X_n>a\})=1$, ...
3
votes
1answer
93 views

Finding value of exponential sum

I'd like to find the value of the following sum $$S(u) = \sum_{n=0}^\infty \frac{e^{iu2^n}}{2^{n+1}}$$ for $u \in \mathbb R$, but I can't seem to do it. Unfruitful work Writing $$S = \sum_{n=0}^\...
3
votes
3answers
445 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: $$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
3
votes
1answer
2k views

Finding the stationary distribution of a markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\mathbb{N}_0$, $q_n >0$ for all $n \in \mathbb{N}_0$ and transition matrix below: \begin{bmatrix} q_0 &...
3
votes
1answer
84 views

Using Borel-Cantelli Lemma to show the almost sure divergence of $S_n/n$

We have independent random variables such that $$\mathbb{P}(X_n=n)=\mathbb{P}(X_n=-n)=\frac{1}{2(n+1)\ln(n+1)}$$ and $$\mathbb{P}(X_n=0)=1-\frac{1}{(n+1)\ln(n+1)}$$ I am trying to show that $\frac{...
3
votes
5answers
188 views

why is $E[E[Y|X]] = E[Y]$

I have a derivation from my book, I have a problem with the very first line: $$ \begin{align} E[E(Y|X)] &= \int_{-\infty}^\infty E(Y|x)f_1(x)dx <- \text{why dx}\\ &= \int_{-\infty}^\infty\...
3
votes
1answer
236 views

Random Variables and Moment Generating functions

Let $(X_i)_{i∈\Bbb{N^+}}$ be a sequence of i.i.d random variables and for $n ∈ \Bbb{N^+}$ set $S_n := \sum _{i=1}^{n} X_i$ and $Y_n := max(X_1, . . . , X_n)$. Assume that the moment generating ...