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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1answer
93 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
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
1answer
58 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
2answers
589 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
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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
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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
2answers
229 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
172 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
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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
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0answers
58 views

Is it reasonable to think of the expectation of an infinite-dimensional vector?

Given a probability space $(\Omega, \mathcal{F}, P)$, a random vector is an $\mathcal{F}$-measurable mapping $X: \Omega \rightarrow \mathbb{R}^{k}: X(\omega) = (X_{1}(\omega), X_{2}(\omega),\ldots,X_{...
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
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1answer
2k views

Understanding conditional independence of two random variables given a third one

I am reading the Wikipedia article on conditional independence. There seems to be Two definitions for conditional independence of Two random variables $X$ and $Y$ given another one $Z$: Two random ...
4
votes
6answers
336 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
4
votes
1answer
48 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
1answer
283 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
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
3answers
715 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
1answer
760 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 ...
4
votes
1answer
501 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
2answers
621 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
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
votes
1answer
153 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
0answers
853 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
4answers
668 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
1answer
170 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
607 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 ...
3
votes
2answers
541 views

Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
3
votes
1answer
159 views

Absolute continuous family of measures

Consider the following family of measures on $(\mathbb R,\mathcal B(\mathbb R))$: $$ K_x(A) = \begin{cases} \int\limits_A \frac{1}{|x|\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy&,\text{ if }x\neq 0, ...
3
votes
1answer
234 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 ...
3
votes
5answers
186 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
3answers
101 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
673 views

Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ ...
3
votes
1answer
79 views

Kolmogorov's sufficient and necessary conditon for SLLN - What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without ...
3
votes
1answer
76 views

Is $E[Z E[Z^2\mid Y] ]$ positive or negative?

Let $Y=X+Z$ where $X$ and $Z$ are independent, zero mean, finite variance r.v. Moreover, $Z$ is Gaussian. Is there are way to say wether \begin{align*} E[Z \ E[Z^2 \mid Y] ] \end{align*} is positive ...
3
votes
0answers
63 views

Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function $\...
3
votes
2answers
183 views

Suggest an example of random variable

Let's consider a probability space $(\Omega, \mathcal{F}, P)$ corresponding to experiments on throwing a dice and defined in the following way: $\Omega = \{1, 2, 3, 4, 5, 6\}$, $\mathcal{F} = \{\Omega,...
3
votes
2answers
89 views

Proof that $A\subseteq B\implies\Bbb P(A) \le\Bbb P(B)$

I have to prove that $A\subseteq B\implies\Bbb P(A) \le\Bbb P(B)$, where $\Bbb P$ is probability. So far I have learned about direct proof, mathematical induction and proof by contraposition. This ...
3
votes
1answer
66 views

probability of a limiting sum

Suppose that $U_i$ are uniformly distributed on (0,1) and are independent. For all possible increasing index sets comprising the family $J$, I am trying to show that $P(\cap_{j\in J} \{\lim_{n \...
3
votes
2answers
131 views

Show that $\int_{\mathbb{R}} F(x) F(dx)=\frac{1}{2}$ if $F$ is a continuous distribution function

If $F$ is a continuous distribution function, prove that \begin{align*} \int_{\mathbb{R}} F(x) F(dx)=\frac{1}{2} \end{align*} What I tried \begin{align*} \int_{\mathbb{R}} F(x) F(dx)&=\int_{\...
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
3answers
247 views

Jumps of independent Lévy processes

Suppose I have two independent Lévy processes $(X_t)_t$ and $(Y_t)_t$, both not continuous. Is anyone familiar and can refer me to a result (or a counterexample) which states that ${\displaystyle \...
3
votes
1answer
268 views

Is a compensated Poisson process uniformly integrable

Let $(N_t)_t$ be a Poisson process with intensity $\lambda$. Define $$ \bar{N}_t = N_t - \lambda t $$ which is clearly a martingale. My question is: is $\bar{N}$ uniformly integrable? I strongly ...
3
votes
3answers
195 views

Relationship “finite mean” <-> "absolutely integrable

What is the relationship between the property of a random variable (i.e. a measurable function defined on some probability space) being absolutely integrable, i.e. $$\mathbb{E}|X|<\infty$$and ...
3
votes
1answer
32 views

Two possible senses of a random variable being a function of another random variable

Given two random variables X and Y (assumed measurable as usual), consider two conditions: There is a (not necessarily measurable) function $f: \mathbb R \to \mathbb R$ such that $Y = f(X)$ holds. ...
3
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0answers
73 views

Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?

I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous. Statement:...
3
votes
1answer
144 views

Basic doubt about stochastic integrals over general local martingales

Consider $M = (M_t)$ is a continuous square integrable local martingale and $$ \mathbb H ^2(M):= \left \{ \psi =(\psi_t)\ \text{is a real previsible process s.t.,} \forall t\geq 0, \ \mathbb E\left \{...
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
74 views

Median of a multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in \{1,2,\...
3
votes
1answer
146 views

Property (ii) of increasing functions in Chung's “A Course in Probability Theory”

I am a bit confused by the line of reasoning on page 2 of Kai Lai Chung's "A Course in Probability Theory". In particular, he is considering a real-finite valued function $f$ which is defined and ...
3
votes
1answer
90 views

Proving (a part of) Hoeffding's lemma

Hoeffding lemma goes like this: *Let $X$ be a scalar variable taking values in an interval $[a,b]$. Then for any $t>0$ $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} (X)}(1+O(t^2\mathbb{V}(X)\exp(O(t(b-a)))...