Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,027 questions
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concentration inequality for averages of dependent random variables
Let $X \in R^n$ be a random vector such that
$$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$
What is a tight bound on
$$P(\sum_{i=1}^n |X_i| > \epsilon)$$
and on
$$P(\max_{1\le i\le n} |X_i| ...
- 461
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0
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Bounding an integral transform ouside a circle (or inside a strip)
Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...
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3
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What is quantum Brownian motion?
It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
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Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
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Distribution of maximum of random walk conditioned to stay positive
I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
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102
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Efficient algorithm for computing the mixed moments of sums of random variables
Let $X_1,\dots,X_m$ be dependent random variables. We are interested in efficient algorithms for computing the following quantity:
$$E\Big[\Big(\sum_{i=1}^m X_i\Big)^k\Big],$$
where $k\in\mathbb{N}$ ...
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many expected streaks imply high probability for a streak
In CLRS, "Intro to Algorithms" section 5.4.3 the following is shown. If a fair coin is flipped n times, the expected number of streaks of consecutive heads of length (1/2)log(n) is $ \Theta (\sqrt n)$...
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134
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Hausdorff distance and sum of independent variables
Consider a probability space $(\Omega, \mathcal{F}, P)$, as well as two sub-$\sigma$-fields $\mathcal{A}$ and $\mathcal{B}$. The Hausdorff pseudo-distance between $\mathcal{A}$ and $\mathcal{B}$ is ...
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Maximal directed crossing of a box using uniform random variables
Take a 1 by 1 box $D \subset \mathbb{R}^2$ and let $U_1,\dots,U_n$ be i.i.d. uniforms in $D$.
Suppose at the start all of $\mathcal{V}_0=\{U_1,\dots,U_n\}$ are viable. At each step pick one of the ...
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Almost sure convergence Banach Space valued Random Variable
Let $B$ be a Banach space. Let $\{Y_{n}\}$ be a sequence of $B$ valued random variables.
Assume
$P(\{Y_{n}\} \mbox{is bounded}) = 1$,
fo every $\epsilon>0$, there exists a finite dimensional ...
user24367
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970
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On average length of sums of unit vectors in R^n
Fix a number m and let us take a set, say A, of unit vectors {v_1,...,v_k} in R^n. Assume that k is large, say exponentially large in n (k=e^{cn}). Let X be the euclidean length of a random sum of m ...
- 2,288
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What is the probability for sequence of lenght L in subset of [n]
I am trying to calculate the probability that i'll have L length sequence in a random subset of [n] when the subset size is k. for example, if n=5, k=4 and L=2 I'll have the below subsets: {2,3,4,5}, {...
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Derivative of a random process
Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t.
I consider applying a (stochastic)derivative operation to the random process. What is the ...
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A (strictly) positively correlated metric space-valued random variables.
Real-valued random variables $X$ and $Y$ are "positively correlated" if $\mathbb{E}[XY]\geq \mathbb{E}X \mathbb{E}Y$, with the intuition that "the larger $X$ is the more probable it is that $Y$ is ...
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If $\mathcal{F}_t$ is separable why is $\mathcal{F}_\infty$ generated by a random variable?
I am reading this introduction to enlargement of filtration and at the beginning of section 2.4 there is a claim that I cannot justify but seems like it should be well known. The author claims that ...
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Estimate on currents in Cayley graphs
Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
- 4,432
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From Brownian Motion to the Heat Equation
Consider a set of N balls that start at the origin. In a given unit of time, $\delta t$, the balls have a probability $p = 0.5$ of jumping a distance $\delta x$ to the right, and the same probability ...
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calculating how much to oversell given an acceptable risk (statistics)
I have a shared resource with a finite capacity (let's say 100), and I have usage data (2 hours average of samples taken each 20 seconds). I accept a risk of 10% per year to reach the capacity.
...
3
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Upper bound concerning Snell envelope
Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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Does every commutative monoid admit a translation-invariant measure?
Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may ...
- 8,512
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2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
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Extension of probability measure from a finite algebra to sigma-algebra with countable many generators
I apologize for probably trivial question, I am far from this field.
If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\...
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Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?
Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic?
If so, what are necessary and sufficient conditions ...
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Expectation of square root of binomial r.v.
Let $X\sim B(n,p)$ denote a binomial random variable. Is there any approximation available for the quantity $E(\sqrt{X})$? Clearly Jensen's inequality holds, but rudimentary tooling around with ...
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2
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789
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Uniform law of large numbers for martingale difference
Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \...
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Are all variables in a set of random variables independent if all pairs are independent?
If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...
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Converse to Girsanov's theorem?
Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
- 1,666
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Markov renewal process with failure?
I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail.
I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = (...
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expected number of cycles in a "random" bipartite directed graph
Consider a "random" bipartite directed graph where (1) on each side, the set of vertices has cardinality n and (2) for each vertex i, we add one (and only one) directed edge i->j at random (drawn ...
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Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
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Can one view the Independent Product in Probability categorially?
One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...
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2
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First hit time in a graph setting
I considered the problem of a form of CTMC evolving in a graph:
Consider a graph of $G(V,E)$ with $|V|=N$ nodes. Each node has a 1-0 CTMC associated with it:
There is a vertex dependent rate $\mu_i$ ...
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Is a measurable homomorphism on a Lie group smooth?
Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...
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Binomial Expectation of Convex Function
Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} \...
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Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
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Exit probability of a Brownian particle.
Perhaps the answer is common folklore among probabilits and stochasticians(!)? But I would like a good lower estimate for the probability that a particle undergoing brownian motion in 1 dimensions ...
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Uncertainty principle in Entropy terms
Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
\int_{\...
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The largest circle that encloses no points on a plane with points placed at $N$ random coordinates
I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [...
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Limit of a Wiener integral
How to show that
$$ \lim_{\alpha \rightarrow \infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$
where $\left (B_s \right)_{...
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Concerning Jump process (Lévy process)
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A \sum_{...
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Eigenvalues of random Hamiltonian matrices
A real $2n\times 2n$ Hamiltonian matrix has the general form
$$H=\begin{pmatrix}
A & B \cr
C & -A^T
\end{pmatrix}
$$
where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
- 1,038
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Is a typical path of a Brownian motion on a torus equidistributed?
Take the usual Brownian motion on $R^d$ and project it to $T^d$, for almost every individual trajectory, will it be equidistributed on the torus? Does this depend on $d$?
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Is there a good concept of a measurable fibration?
In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability ...
- 8,512
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votes
1
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Taking the partial derivative of the t-CDF with respect to the degrees of freedom
I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in ...
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1
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201
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Moments of random matrices - when are they finite
I need to evaluate the moment
$$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is
$$X=ZZ^{\ast},$$ where
$Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...
25
votes
6
answers
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Metrization of weak convergence of signed measures
Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
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votes
4
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Probability that randomly chosen integers from a restricted set of natural numbers are coprime
We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is
$$
P(k) = \frac{1}{\zeta(k)}.
$$
I am looking at a special case of ...
- 5,470
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0
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243
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Factorization of permutations.
Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Consider the following ...
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Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 2,201
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Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
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