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.
9,023 questions
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Which sub-sequence selection rules preserve the iid property?
Let $\xi_1,\ldots,\xi_n$ be an iid sequence of random variables. If we take a sub-sequence $\xi_{i_1},\ldots,\xi_{i_k}$ with constant indices $1\leq i_1 <\ldots <i_k\leq n$, then the sub-...
- 251
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68
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What can be said about moments of probability distribution if it satisifies hemholtz equation?
From physical considerations I have observed, that probability density in region of interest satisfies
$$
\Delta u(x) + \phi(x)u(x) = f(x),
$$
where $\phi(x)$ and $f(x)$ are both given functions and $...
- 355
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102
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Probability of random variable being lesser than the other
Say there are two independent random variables, $X$ and $Y$, and we have samples $\{x_1,\dots x_n\},\{y_1,\dots y_n\}$. I am interested in bounding the probability of the event $C = \mathbb{1}_{X<Y}...
- 197
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34
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What kind of prior on edge existence would form graphs that are unions of complete (sub)graphs?
Suppose a graph has $n$ vertices.
First question: is it possible to give a (nontrivial) prior probability on edge existence so that if a graph is created by querying the prior on the $\binom{n}{2}$ ...
- 118
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82
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The effect of channel error on the determinant of transmitted matrix
Assume the following matrix
$$
E:=\left(
\begin{array}{ccccc}
e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\
e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\
\...
- 313
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96
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How to get some information about a random variable if we know very little about its distribution
Suppose that $X, Y$ are random variables,both from a probability space to $(0,\infty]$, such that X and $1+Y$ have the same distribution, $Y=Z_1$ with probability equals half, otherwise $Y= \frac{...
- 228
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87
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Variation on stones in buckets
This is a spinoff, see Collecting stones in n buckets. Frankly speaking my only motivation is that I became curious: what happens if one redistributes the stones into the same buckets?
More ...
- 18.4k
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54
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A multifractal model of asset returns - Mandelbrot, scaling result
I am looking at the paper in the title and I am trying to derive their result in equation $2$. Here is what I obtain. Start with:
$$X(ct) \stackrel{d}{=} M(c)X(t)$$
where $M(.)$ and $X(.)$ are ...
- 101
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236
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Laplace transform (or characteristic functional) of atomic random measure
A random (nonnegative Radon) measure $M$ (on $\mathbb R^n$, say) has its law characterized by the Laplace transform $\mathbb E\exp(-\int \varphi(x)\ M(dx))$, $\varphi\in C_c^+(\mathbb R^n)$ (...
- 3,085
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86
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Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$
In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...
- 10.4k
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167
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Finding the right σ-algebra. Question on uncertainty related to the secretary problem
Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...
- 25
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57
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Where can I find this article of Doléans-Dade?
I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...
- 630
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252
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Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
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81
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Prokhorov convergence of Gaussian measures
Consider a Hilbert space $\mathcal{H}$ and a sequence of centered Gaussian measures $\mu_n$ on it. The covariance operators of $\mu_n$ are defined via their eigenpair(eigenbasis and eigenvalue)) as ...
- 157
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51
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derivation of a gap related to extreme value theory
I have an expression to evaluate as follow:
$\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$
where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows:
\...
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124
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Maximal inequality for Markov process
For a Markov process $\{X_n\}$ is there any inequality available for
$$ E[\sup_{0 \leq n \leq k} X_{n}]$$
in terms of moments of $E[X_n], 0 \leq n \leq k$
- 317
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88
views
CLT for sums of an infinite sequence of rv with an asymptotic distribution
Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
- 171
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160
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How to check numerically iterated logarithm law ? (How to choose cutOff lim_n sup_{m: n<= m<= CutOff} ) ?
The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then
$$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...
- 24.9k
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1
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150
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Weak convergence of process
Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) \...
- 243
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107
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Conditional version of martingale difference concentration inequality
Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following
$$P(|M_{n+1}| > u|\...
- 317
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453
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Integral involving modified bessel function of second kind, exponential and power
I need to compute the following integral.
$$
\int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,.
$$
where $$ a>0$$
and $b$ can be greater than zero or less than zero but it is not a complex ...
- 119
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322
views
Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
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1
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360
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Weak existence for modified Tanaka SDE
Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE
$dX_t = dW_t + dL_t^0(X_t)$,
where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$....
- 43
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1
answer
244
views
Weak solutions of linear parabolic PDEs and corresponding SDEs
It is well known that for an Stochastic differential equation (on the real line) of the form:
$dX_t = \mu(X_t)dt + \sigma(X_t)dW$
where $W$ is the standard Wiener process, the transition probability ...
- 3
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216
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Hoeffding's lemma for unbounded r.v with bounded exponential map
Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: $$E[...
- 9
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168
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A path optimisation problem
Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...
- 367
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81
views
Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
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44
views
Is it possible to estimate the Interaction information of three variables without knowing their joint distributions?
I want to have a measure of the "synergy" between two players in a game. Each player has its own win ratio (won/played), which I'm modeling as two binomial distributed random variables X and Y. A ...
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260
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Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
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186
views
Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
user76167
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81
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An asymptotic set containment problem [closed]
Given a set of $n\in\Bbb N$ integers $\mathcal S$, suppose we choose two sets:
$$\mathcal S_{\mathsf{small}}\subseteq\mathcal S$$
$$\mathcal S_{\mathsf{big}}\subseteq\mathcal S$$ with cardinalities
...
user76479
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0
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355
views
Summing up costs over a Markov chain
I apologize in advance if this question is too simplistic to be appropriate for MathOverflow. I have inquired in multiple places but have found little to indicate that this is a previously studied ...
- 101
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76
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What is the success probability of this stochastic process?
Suppose you have $k$ black balls and $X\cdot k$ white balls.
The procedure start with you having a bag containing $y\le k$ white balls (e.g. $k+1,\ldots k+y$).
In every iteration:
A single white ...
- 1
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2k
views
Probability two random intervals overlap
I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows:
Given N randomly ...
- 101
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194
views
Derivative of a cdf with respect to a parameter
Given two independent Random Variables $X$ and $Y$ with known distributions, I would like to know if I can say that the expression
$$
\operatorname{Pr}( f (t'+Y-X)+Y-X < z)
$$
is increasing in ...
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117
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Ergodicity property for continuous-time Harris positive Markov process
I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If $\Phi$ ...
- 109
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104
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Why is this distribution exponential?
Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...
- 193
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165
views
Expected length of minimum spanning trees
For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...
- 1
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74
views
conditionning by a Gaussian field
I know that if $(X,Y)$ is a Gaussian vector, then $(X|Y=y)$ is a Gaussian vector which covariance matrix is explicit in function of the covariance matrix of $(X,Y)$, and does not depend on $y$.
What ...
- 1,352
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320
views
Gromov-Hausdorff distance measure between minimum spanning trees
I am trying to compare minimum spanning trees through time. I have two questions:
1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
- 1
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119
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Estimating the number of colors in a bucket
This question was previously posted to Math Stack Exchange here.
Suppose we have a bucket containing a large (but known) number of balls. Each of the balls has a color. We don't know how many colors ...
- 1,302
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444
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How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
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0
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583
views
When an integral with respect to a Poisson point process is finite?
Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ \...
- 724
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145
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Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
- 215
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216
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Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903
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133
views
What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
- 19
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42
views
Probability of close approach for multivariate normal variables
The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
- 195
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1
answer
208
views
Lower bound for median of independent outcomes
Consider a stochastic variable $X$ taking positive real values and the events $P(X\geq a)\leq\frac{1}{3}$ and $P(X \leq b) \leq \frac{1}{2.9}$. We define $X_m$ as the median of $k$ independent ...
- 75
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454
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Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
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0
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57
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Numerical method for self-consistency of one-dimensional probability density function
I have an integral equation for self-consistency of one-dimensional probability density function, like this
$$\rho_x(x) = \frac{1}{|a|}\int \int \rho_x\left(\frac{s-b}{a}\right) \rho_P(p) \delta(x-g(...
- 111