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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62 views
A basic doubt on CDF and pdf
Let $X$ be a random variable and $A$ be a Borel subset of $R$.
Then $$P(X \in A) = \int _{A}f_X(x) dx$$
where $f_X$ is the probability density function of the random variable. Now I have not ...
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62 views
How does this violate probability theory?
Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$)
$p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$
$p(Y = -1) = .5$, $p(Y = 3) = .5$
Question: Despite not being handed any information ...
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26 views
Area of a Random Polygon
The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
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26 views
Haar system and martingale
Today our lecturer used martingale theory to show that the Haar system is a basis for $L_1[0,1]$ (we're operating on the probability space $([0,1],\mathcal{B},P)$, where $P$ is the Lebesgue measure). ...
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24 views
limit distribution of possion distribution
Assume Xn is possion with mean $\lambda_n$ and suppose that $\lambda_n\rightarrow\infty$ as $n\rightarrow \infty$. Then how to show Xn is AN$(\lambda_n,\lambda_n)$. I've tried to use characteristic ...
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28 views
Another homework question in ergodic theory
Again the source is http://www.math.ucla.edu/~biskup/275c.1.13s/PDFs/HW1.pdf this time I'm looking at #6 the part that is left as an open-ended question. If $f \in L^1$ and $\phi$ is a measure ...
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37 views
Homework questions in ergodic theory
Let $X_1, X_2, ...$ be iid. If $f: \mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ is measurable wrt the product structure it's $L^1$ under the distribution measure induced by the $X_i$ then why is it ...
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40 views
Almost Surely Convergence
I need some help with computing the lim inf and lim sup of $ \frac 1n \sum_i X_n$ where the density of variable $X_n$ is absolute continuous, say, f(x) = exp(-x). I am interested in using the ...
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32 views
Maximum likelihood in exponential family: $\partial_{\theta,\theta} \ln L = -\mathrm{Var}(T)$, $T$ sufficient for $\theta$
I'm confused on how the second derivative of the log-likelihood is computed in an exponential family.
There is a result which says that
If $T=T(X)$ is the natural sufficient statistic for ...
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21 views
Suitable change of measure with importance sampling
I'm working on a project on Importance Sampling. When one tries to estimate the small probability $\alpha:=\mathbb{P}(X \in A)$, one can do a change of measure. There is change of measure which ...
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30 views
Brownian motion first hitting time for $0$
Let's say that I have a family $(X_t)_{t\in [0, \infty)}$ that is continuous in t everywhere. (I do not mean almost everywhere, because then there'd be funny but perhaps easily resolved measure ...
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0answers
70 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
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0answers
20 views
Is the converse of this convergence theorem about characteristic functions true?
In Kai Lai Chung's A Course in Probability Theory,
Theorem 6.3.1 says:
Then he provides Theorem 6.3.2, which is not included in this post,
and then gives a corollary of Theorem 6.3.2,
I was ...
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0answers
27 views
Stopping time and expectations
Let $(X_n)_{n\geq 0}$ denote a martingale with respect to some filtration $(\mathcal{F}_n)$, and let $\tau$ denote a stopping time with $P(\tau<\infty)=1$. Assume that there exists an $M$ such that ...
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33 views
Non-independent two consecutive draws from two urns
Suppose there are two urns: in urn A, there are r red balls and w white balls. In urn B, there are b black balls.
Suppose we do the following experiment: draw k balls from urn A. Among those k balls, ...
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26 views
canonical form of dyadic martingales
Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$.
Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
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0answers
31 views
Continuous random sampling with replacement.
Construct a set $s\subseteq[0,1]$ by sampling points in $[0,1]$ with uniform probability density $x\leq1$ so that $|s|=x$. Interpret this as a sampling frame during which data is captured. Now, ...
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0answers
36 views
Stuck on proof of a martingale equality (similar to doob's inequality)
The question is: Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. Prove that for every $x > 0$
$$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} = 1\wedge ...
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0answers
21 views
Estimate on Galton-Watson process distribution
Let $(Zn)_{n\in \mathbb N_0}$ be a Galton-Watson process, i.e.
$$ Z_{n+1} = \sum_{k=1}^{Z_n}\xi_{n,k},\qquad (\xi_{n,k})_{n\in \mathbb N_0,k \in \mathbb N} \quad \text{i.i.d } \mathbb N_0 \text{ ...
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86 views
Change of probability measure and a continuous-time Markov chain
Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
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32 views
Cylindrical sigma algebra answers countable questions only.
I got a missing link in some in the following (standard) textbook question:
Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
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34 views
Upper bound on truncation error of a fourier series approximation of a pdf?
Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation:
$$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$
...
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32 views
convergence of discrete random variables with finite entropy
Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
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35 views
Furstenberg recurrence
I am having trouble verifying exercise 1 here. Can I get some hints/solutions? (It's not homework, I am just reading up on it for my own interest.)
Theorem 2. (Furstenberg multiple recurrence ...
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16 views
Analytic random function
Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function.
I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic.
What are the minimal conditions needed? ...
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38 views
Expectation of Random Variables - Measure Theory
I am trying to do the exercise 2 of section 3.2 of the book "A Course in Probability Theory by Kai Lai Chung". Problem asks to show:
If $\mathscr E(\left|X\right|)<\infty$ and $\lim_{n\to ...
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61 views
Probability distribution with maximal entropy on $[0,1] \cup \{2\}$
For given closed set $F$ on $\mathbb R$ one can think of probability distribution $\mathbb P^\ast_F$ with support on $F$ and with maximal entropy. It is well known that
If $F=[0,1]$ then $\mathbb ...
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43 views
asymptotic order of the variance of the maximum of iid standard Gaussian
Suppose that $X_1,\cdots,X_n$ are iid standard Gaussian. $X_{(n)}$ is the maximum of $(X_1,\cdots,X_n)$, how can I find the asymptotic order of $VAR[X_{(n)}]$?
The density function of $X_{(n)}$ can ...
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57 views
sequence with almost surely convergence & moment of order 2
Let's say we have iid random variables $(X_n)$ s.t. $X_1$ admits a moment of order 2.
For $n \geq 1$, does the following sequence converge almost surely or not? Why? And how to see/show?
$$A_n = ...
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39 views
What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?
It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
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44 views
Need reference for layer-cake representation of conditional expectation
I'm looking for a reference to a conditional version of the layer-cake representation, i.e.,
$$
...
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26 views
Is open-ball-weak convergence of borel-measurable random elements the same as borel-weak-convergence?
The definitions are from David Pollard, Convergence of Stochastic Processes, IV.1.1 p.65
Let $(\Omega,\mathcal{A},P)$ be any probability space.
Let $(S,\mathcal{S})$ be a metric space with any ...
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26 views
The field generated by the empirical distribution function
The Gist
I've got three question related to the field generated by the empirical distribution function ("the empirical field").
Is the empirical field identical to the symmetric field?
Are all ...
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45 views
Showing a certain process has $\limsup X_t$ bounded almost surely.
This question has been solved.
I'm working on a problem where I need to show
$$\limsup_{t \rightarrow \infty} X_t \leq \sqrt{c}\quad \text{a.s.}$$
where $X_t$, $t \geq 0$ is a stochastic process ...
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0answers
74 views
Borel-Cantelli lemma proof $\text{P}\left(\lim_{n\to\infty}{ M_n\over{\log n}}=\frac{1}{\lambda}\right)=1$
Show $\displaystyle \text{P}\left(\lim_{n\to\infty} {M_n\over{\log n}}=\frac{1}{\lambda}\right)=1$where $M_n=\max_{1\leq k\leq n}X_k$ and $X_k \text{~ Exp}(\lambda) $
! All I have so far is (which ...
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0answers
29 views
Exchanging LimSup and Sup
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$.
Consider a bounded upper semicontinuous function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$ is ...
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38 views
Uniform Integrability is Iterative?
Consider a probability measure $m(\cdot)$ on $Y \subseteq \mathbb{R}^m$, so that $m(Y) = 1$.
Let $X \subset \mathbb{R}^n$ be a compact set, and $f: X \times Y \rightarrow X$ a locally bounded ...
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0answers
68 views
Large Deviations Problem
Let $\left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\left(\Omega,\mathcal A, \mathbb P\right)$, $X_1$ with mean $\mu$, and
$$
L(\lambda) =
\begin{cases}
\log\mathbb E\left(e^{\lambda ...
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53 views
Definition of $x\left<Y\right>$ notation in probability theory
I am working on the basics of probability theory in Koller's Probabilistic Graphical Models - Principles and Techniques. Unfortunately I am having trouble understanding a formal definition (possibly ...
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40 views
Cramer-Rao bound for $\chi^2$ distribution parameter estimates.
I've stuck in unpleasant problem with noncentral $\chi^2$ distribution.
I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
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42 views
Expectation related to renewal measure
Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely ...
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0answers
45 views
Separable processes
If $X_t$ is a separable (Gaussian) process, is the process defined by
$Y_0=0, Y_h= \frac{\lvert X_{t+h}-X_t \rvert}{|h|^\gamma} $ for $h>0$, and $-t<h<0$ and some fixed $\gamma>0$ also ...
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0answers
55 views
What's the probability of creating a Hello World Program?
Consider an application that has knowledge of all characters on the keyboard i.e. if asked to do so it can randomly choose any character and output it.
Now, in the programming language Java consider ...
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0answers
59 views
Reference request - maximum conservative extension of a probability assignment
I made up some definitions which probably lead to some interesting mathematics. However, I suspect they've studied before. So that I don't end up reinventing the wheel (always bad!), I'm after a ...
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45 views
how can I calculate the mean and variance of random variables?
There are two dependent (correlated) random variables with Rice (Rician) distributions $R_1\sim (u_1,s_1)$ and $R_2\sim(u_2,s_2)$. Then, how can I calculate the mean and variance of random variable ...
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31 views
How to make this inference: Degree of a node in a graph is significantly diffenrent from poisson distribution
I am working on Gene-Gene interaction graphs. I build a graph by adding edges between genes (nodes) which show statistical interaction in predicting a quantitative parameter value (say, brain volume) ...
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0answers
21 views
Initial jump of semimartingale integrator
I've read that it is a common assumption in the literature on stochastic integration that a semimartingale integrand $S$ may jump at $t = 0$ and a common convention is to assume $S_{0-} = 0$, so the ...
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0answers
73 views
Central limit theorem - speed of convergence in center vs tails
I've been told that one of the implications of the central limit theorem is that as we increase the sampling of random variables, we converge faster to a normal distribution in the center and slower ...
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0answers
34 views
Independence of Brownian motion-related stopping times
Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
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54 views
Likelihod ratio test: $f_0(x)=2x$ vs $f_1(x)=3x^2$ : $2n$ degree of freedom?
Let $\left\{ X_i \right \}$ be an $n$-sample with pdf $f$.
Show that the likelihood ratio test statistic for
\begin{align}
H_0 &: f(x) = 2x \\
H_1 &: f(x)= 3x^2
\end{align}
has a $\chi^2$ ...
