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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Spanning subgaph with trivial Poisson boundaries
Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-...
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Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
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How does changing the transition probabilities affect the concentration of a position-dependent random walk?
Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all $...
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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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Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain
Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique (=...
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Central Limit Theorem (and Berry-Esseen theorem) for non-independent variables
Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties:
For any given $1 \le L \le n$, all
subsets of
$(X_{n,1},\...
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Central Limit Theorem for additive function of permutations of sequences
Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...
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Binomial moments for the number of events occuring
Let $A_1,A_2,\dots,A_n$ be events on a probability space. For $0 \leq k \leq n$ let
\begin{equation*}
S_k=\sum_{1 \le {i_1}<{i_2}<\cdots<{i_k} \leq n} P(A_{i_1} \cap \cdots \cap A_{i_k}).
\...
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Is the Binomial Expectation of Convex Function Convex in p?
Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
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Probability theory over noncommutative ring? [closed]
Observation: Entropy is a metric over some non commutative ring. Indeed, if we exponentiate the standard entropy definition
$\displaystyle H(X) = -\sum_{x \in \mathcal{X}} p(x) \ln p(x).$
we'll get
...
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What is the order of the lower tail of a Chi-Squared distribution?
Let X be a random variable with having a chi-squared distribution with n degrees of freedom and let y be some real number at most n. Is it known how P (X < y) behaves at least in some reasonable ...
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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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Estimates for the mixing time of a Markov Chain with biased initiation
Imagine I have some Markov process consisting of a biased random walk on the integers, over some interval $[0, L]$, with $+1$ and $-1$ step probabilities of $p$ and $q$, respectively, s.t. $(p + q) = ...
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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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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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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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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 ...
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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 ...
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552
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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 ...
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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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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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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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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.
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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 ...
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Heuristically false conjectures
I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
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Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
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Iterating Random Matrix Operations
Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...
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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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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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Probability distribution for the size of an ordered set of (randomly pruned) integer pairs with intersection constraints on successive elements in the permutation
Update: To write a quick preamble, this question is basically asking that, if you take all possible pairs of some set of characters, call these pairs elements of the set $S$, and if you throw out some ...
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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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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 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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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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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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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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An elementary probability question
Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.
Consider $n+1$ samples $X_0, \ldots, X_n ...
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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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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 ...
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Is there a general process for conditioning a stochastic process above a boundary?
$(X_t, Y_t)$ is a two-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. Given its transition function $a(x, y | x', y')$, I would like to condition the process on $\inf_{s \...
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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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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 Bruss-Yor conjecture about an iterated integral
Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$?
This is a conjecture of F. ...
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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
$$ \nu \left( dx\right) = A \sum_{...
- 710
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invertibility of a matrix with a Gaussian perturbation
Suppose that $A$ is an arbitrary fixed $n\times n$ matrix and $G$ a random $n\times n$ matrix with i.i.d. $N(0,1)$ entries. Is there a simple proof that $A+G$ is invertible with probability 1?
What ...
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Is a typical path of a Brownian motion on a torus equidistributed?
Take the usual Brownian motion on $R^d$ and project it to $T^d$, for almost every individual trajectory, will it be equidistributed on the torus? Does this depend on $d$?
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Moments of random matrices - when are they finite
I need to evaluate the moment
$$\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 ...
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