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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Limit of the stochastic process at time 0
This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process,...
- 153
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random matrix products reference
For a long time the standard (though not the easiest to find) reference on random matrix products was Bougerol and Lacrois:
Bougerol, Philippe, and Jean Lacroix. Products of random matrices with ...
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Deciding whether or not an exponentially distributed random variable exists in a set via the use of a "black box" function
I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ......
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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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2
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305
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Properties of the Euler Discretization of a diffusion
Let $X$ be a continuous 1-d diffusion:
$$
dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x.
$$
W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties.
Let $Z^n_1,Z^...
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1
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First order approximation of the current in ASEP
I am searching for an elementary proof of the first order approximation of the current of particles in ASEP (asymmetric simple exclusion process) and TASEP (totally asymetric). To avoid technical ...
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258
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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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Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid
Consider a simple random walk on the lattice $\mathbb Z^2$ starting at the origin $(0,0)$ where in each step, one of the four adjacent vertices in chosen uniformly at random, i.e. with probability $1/...
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"Nice" eigenvectors for (square of) adjacency matrix of a bipartite graph?
Let $G$ be a bipartite graph, and let $A$ be its adjacency matrix.
I was wondering in this case whether $A^2$ will have nice eigenvectors that reflect combinatorial structure of the graph. I'd be ...
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Computing a density function for the integral of a stochastic process, given its transition function
$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
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Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...
4
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2
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675
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between "giant-component" and "fully connected"
This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giant-component" and "fully connected"?
e.g. a detailed picture for say, $p_n=\frac{(...
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Algorithm for numerically approximating the Prokhorov metric?
Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures?
Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
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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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Find a minimum entropy code for a simple gibbs random field.
Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about.
In the rest of this ...
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Median-of-k elements
Hello,
Assume I am given a sequence of $n$ elements (by sequence I mean an ordered set). I want to randomly pick $k$ elements out of these $n$ elements, where $k$ is an odd number $\leq n$.
Then out ...
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289
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Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
- 61
1
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1
answer
142
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Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
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What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
For example for a text which is generated by a multinomial distribution over words, ...
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585
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Existence of a probability measure with "confined" zero measure sets
Hi, I am struggling with the following question that is tangentially arising from a paper I'm working on. It is not at all essential for the revision but it would be nice to know if there is a ...
- 900
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3
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991
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Berry Esseen inequality for multidimensional distributions
The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary
$$
\sup_{t \in \...
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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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Bochner integral of stochastic process = path by path Lebesgue integral?
After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
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8
answers
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A sudden smiley? :-)
This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...
- 151k
3
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251
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Permutations & Balanced Distribution
I would like to implement a form of consistent hashing using a set of permutations.
The rules are as follows:
I have Y=~32 buckets and X items. Buckets may be "alive" or "dead". Items are to be ...
- 31
13
votes
1
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869
views
Lotteries, Turan's problem, and minimization of risk
Suppose I am a high-volume broker aiming to make some money on a state lottery. In this lottery, six balls are drawn from a population of (let's say) 50, without replacement. A ticket is a choice of ...
- 19.2k
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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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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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2
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631
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Proving that a complicated function is eventually concave
I have a function $f:\mathbb{R}^+ \to \mathbb{R}^+$ that I want to prove is eventually concave - i.e. that there exists $\gamma _0 > 0$ such that for every $\gamma>\gamma_0$, $f(\gamma)$ is ...
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Enumeration of quadrangulations with a boundary and simple faces.
I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct.
...
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1
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516
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Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane
Imagine I perform the following procedure:
[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.
[2] At time point $t_2$, I center a ...
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Covariance of INID order statistics [closed]
In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...
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contraction property for conditioned SDEs
Consider a strongly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (*)$$ where $W$ is a standard Brownian motion. If $(X_t)_{t \geq 0}$ ...
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Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
- 5,502
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0
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Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
- 141
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2
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Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.
Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ).
I have a ...
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Approximation of stochastic differential equations
Consider the two following real Stochastic Differential Equations (SDE) starting from the same initial condition:
$$dx_t = f(x_t)dt + \sigma dB_t$$
$$dy_t = f(y_t)g_{\epsilon}(y_t)dt + \sigma dB_t$$
...
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If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
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715
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What time does it take for irrational rotations to hit an interval?
Hi,
Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take ...
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332
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Expectation where linearity does not hold
We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude ...
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2
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315
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Sampling from a recursively defined distribution
I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...
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429
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Ising model on a cycle
The Ising model on $\mathbb{Z} / 2d\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d-1}) \in \{-1,+1\}^{2d}$ a probability proportional to $\exp\\big(\beta \sum_i x_ix_{i+1} \\big)$. The ...
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3
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Concentration results for inner products of two independent random gaussian vectors
Hi,
I wanted to know if there are standard results on concentration of absolute
value of inner products of two random vectors. Thus if $X, Y \in R^m$ are two
independent random vectors with each ...
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3
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1k
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what is the probability that a scissor became the champion?
Here is a question from one of my students:
suppose 8 players are in an elimination match. The players are marked with marked with either R (for rock), P (for paper) or S (for scissors). If two ...
- 201
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1
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368
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Product of probability densities of the form x^{-t} exp (-ax)
I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>...
5
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2
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1k
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Derivatives through random variables?
Suppose I have some random variable X with probability distribution P(.;theta). Suppose I have a single sample x from this distribution.
Does it make any sense to ask for the derivative of x with ...
- 173
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1
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294
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Kalman Filter...Denoising measurement data to track objects
Hi Everyone,
I am about to implement a Kalman Filter in a software.
I found this very helpful article here:
http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx
The example helps a lot, ...
- 11
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0
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104
views
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 ...
- 21
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2
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540
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Maximum entropy priors in infinite dimensional spaces
Is there an extension of maximum entropy probability distributions for function spaces?
For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-...
- 1,160
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712
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bound the hitting time in Markov chain
Given a finite-state Markov chain $M$ and assume there is a terminate state $f$ which is reached with prob. 1, I am interested in the distribution of hitting time $T$ of $f$, namely, $T$ is defined as ...
- 41