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.
9,022 questions
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question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
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Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
- 585
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215
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Residual lifetime of heavy-tailed random variable
The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula
\begin{equation}R(t)=P[X_\text{res}\leq t] = 1-\frac{1}{\mathbb{E}[X]}\int_{y=0}^\...
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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Dominating Poisson with parameter depending on a Bernoulli
Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value ...
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204
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Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators
Consider the operator on $\ell^2(\mathbb{Z})$
$$
H = \Delta + v.
$$ Here $\Delta$ is the nearest neighbour Laplacian on $\mathbb{Z}$, $\Delta_{k, \ell} =1 $ if $|k - \ell| =1 $ and zero otherwise, ...
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134
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Mittag-Leffler function and Laplace Integral
Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function.
I am looking for a full proof of the following fact (a reference to a proof in the ...
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257
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Sum over a product of binomial coefficients related to a collision problem
I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:
$$R\left(n,m,j\right)=\sum_{k=0}^{n}...
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408
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Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
- 1,109
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522
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How to simplify conditional probability of union of several events
I have an output binary scalar, $y∈B=\{0,1\}$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=\{0,1\}$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
- 133
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matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
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404
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When is the median closest nearest-neighbor distance larger than the mean closest nearest-neighbor distance?
Consider a random Poisson process in an $d$-dimensional cube of arbitrary size (alternatively, consider an arbitrarily large $(d-1)$-dimensional sphere in an $d$-dimensional space). If we have a ...
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208
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Brownian particles in a box: the probability that a sphere (of some radius) centered on a particle only contains one particle for a duration of time
Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a ...
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104
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Minimum distance larger than a fraction $f$ of the closest nearest-neighbor distances for points placed by a random Poisson process?
Consider a random Poisson process on arbitrarily large volume in $R^d$ enclosed by an $R^{(d-1)}$ dimensional sphere. The process terminates when a density of points $\rho$ is achieved (letting $N$ ...
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213
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Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum
Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...
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Can I obtain the limit value of a linear spectral statistics using Stieltjes transform?
I would like to calculate the limit value of a linear functional
\begin{equation}
\lim_{n\rightarrow\infty}\mathcal{I}_n=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n f(\lambda_i)=\lim_{n\...
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145
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multivariate integral calculation in closed form
I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
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151
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Inequality relating stationary probabilities and transition probabilities
Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains ...
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Measure induced by function
It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-...
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probability P that all circles are connected with each other.
Let N circles with homogeneous radius r are deployed with Poisson distribution in area A. These circles are connected if there euclidean distance is less than r.what is the probability P that all ...
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Finding conditions on unspecified CDF that permit a solution to an equation
[A duplicate thread can also be found at
https://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]
Let $F(\alpha) := \mathbb{P}(\...
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347
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An interesting version of the problem “balls into bins”
Consider n people, each has k identical balls. Each people choose k different bins from m bins, constrained by the condition that there are no two people choose exactly the same k bins. For instance, ...
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91
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Pruning copies of an element from a multiset via a uniform random selection process - does vigilance matter?
This is an extension of a previous question of mine (nicely answered by Douglas Zare): Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
Say I fill a ...
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118
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mathematical expectation of length of dependency well.
We have these assumptions:
$W$ is a finite set
$\mathcal W$ is the set of all functions $f:W\to \mathcal P(W)$.
$p:\mathcal P(\mathcal W)\to [0,1]$ is a probability measure.
For each $w\in W$ and $m\...
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229
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Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
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160
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
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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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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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656
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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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112
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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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Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral
Hi, I have the following expected value to compute
$E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...
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Probability of summing products of irreducible polynomials in a finite field to zero
Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
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155
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Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
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124
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Is there a known asymptotic scaling for the probability of recurrence for a walk on $Z^d$?
I'm curious if there is a known asymptotic scaling for the return-to-origin (i.e. recurrence) probability for a random on $Z^d$ as a function of $d$?
Mathworld gives the recurrence probability:
...
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179
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Arithmetic properties of erf functions
I was messing around with Benford's law trying a proof to fill up time on a Saturday, and I ran into a problem. I have the equation $\frac{\mathrm{erf}(2x)-\mathrm{erf}(x)}{\mathrm{erf}(10x)-\mathrm{...
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166
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Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
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coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
- 295
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Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative
Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
$P(x(t))=\frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp(-\frac{(...
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145
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Finding expectation of size of a subgraph
I have been trying to implement a algorithm but got stuck in finding expectation of the size of the subgraph.
n - size of the network.
d - at most number of communities a node could participate ...
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165
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transition probability convergence for Harris chains - Durrett.
Dear mathoverflow.
This is a question to a proof in a graduate text. I have asked two professors at my university without help, so I hope it suffices in difficulty for this forum otherwise I ...
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303
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hitting probability for integrated Ornstein-Uhlenbeck process
Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...
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104
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Proving that a property holds for random sequences with given marginal distribution by rearrangement
I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
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107
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Can one combine (join) probabilities from 2 aspects of a related process?
Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...
- 111
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135
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expected number of shared 1s between two binary strings from a given set
Let say, I have two binary strings with length N, chosen from a set where there are $2^N-K,(K \ge 0)$ independent strings. What would be the expected number of Ones at the same index from two randomly ...
- 21
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283
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Probability and events [closed]
Hi everyone
The question is the following:
A certain event may or may not take place. So we say that if we focus on it one time, it has a probability p of being satisfied (0 <= p < 1)
If we ...
- 3
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111
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Stationarity of an Integral Process
Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
$$...
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1
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182
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How to Rigorize an inequalities argument
Context
I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property.
What I need to prove:
There exists some constant $c$, and functions $p,...
- 3
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352
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prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra
I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.)
Taking the conditional expectation of an RV with respect to ...
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161
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T. Lyons Criterion
Hello all,
I want to prove that any flow on the following tree must have an infinite energy.
The structure of the graph is (taken from R.Lyons and Y.Peres book)
"We’ll construct a tree $T$ embedded ...
- 1
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130
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span of symmetrically truncated symmetric random variables
If $X_i$ are symmetric independent random variables, is $\vert \sum X_i I_{\vert X_i \vert < N_i}\vert $ stochastically smaller than $\vert \sum X_i \vert$ ? Is it comparable in any way which ...
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