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.
9,024 questions
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Puzzle in Martin Gardner book [closed]
What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? ...
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Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
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expected number of balls in k emptiest bins
My problem is the following: throw (randomly, independently) $p$ balls in $n$ bins. What is the expected number of balls in the $k$ emptiest bins?
I have some results about the expected number of ...
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Stochastic Integrals and Cauchy Variables
I hope there is a straighforward literature-pointer here.
If I were interested in $\sum_{t=1}^{n} f(t) X_{t}$, where $X_{t}$ consists of independent normal random variables, I could approximate the ...
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Weak*-continuity of regular conditional probabilities "in time"
Let $(\Omega, F, (F_t)_{t\geq 0}, \mathbb{P})$ assume that $(X_t)_{t\leq T} $ is some cadlag, real valued stochastic process, not too bad: say something like a Brownian Motion and some Poisson finite ...
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Did Joseph Doob prove that random sequences don't exist?
In the book "The Mathematical Experience" it says:
"An infinite [binary] sequence $x_1, x_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x_{n_1}, x_{n_2}, \ldots$...
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Ping-pong relief map of a given function z=f(x,y)
I have an idea to design a type of
Galton's Board
to "draw" a relief map of a given two-dimensional function $z=f(x,y)$.
A typical Galton's Board drops, say, ping-pong balls through a series
of evenly ...
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Probability that a certain Markov process has produced a given state
I am looking for advice on the following practical problem. Please keep in mind that this came up in a practical application.
In the context of Markov chains, we have $N$ states, with $N$ very large....
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Number of real solutions of a random equation
Let $(J_{ij})$ be an $n \times n$ random matrix with i.i.d Gaussian centered coefficients with $\displaystyle \mathbb{E}[J_{ij}^2] = \frac{\sigma^2}{n}$.
Let the random variable $A_n(\sigma)$ defined ...
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Is there a problem with the Wishart distribution?
It seems that the Wishart distribution scales problematically with dimension. I start with some background, which should make the question reasonably understandable to a general math and statistics ...
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name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$
Given a real-valued data set $ x_1, \dots, x_n $, what do you call the quantity
$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$$
This seems like a pretty basic ...
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Diffusion convergence
Consider diffusion:$$d\eta^x_{t}=\sigma(\eta^x_{t})dW_{t}+\mu(\eta^x_{t})dt,\quad \eta_0^x =x,$$ where $W$ is a Wiener process. We assume that $\sigma,\mu$ are such that the diffusion is well-...
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Reading Material on Couplings
Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?
I'm working on a research project on Poisson approximations and am looking to perform a coupling ...
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Who is the weak* sequential closure of the set of finitely supported measures on the integers?
Let $X$ be a topological space and $Y\subseteq X$, the sequential closure of $Y$ is the set of elements in $X$ that are limit of sequences belonging to $Y$.
Let $\mathcal M_{\text{fin}}(\mathbb Z)$ ...
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law of iterated logrithm
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is
a standard Brownian motion. By law of iterated logarithm, one has
$...
- 1,399
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Probability of returning to starting point before hitting adjacent point for RW on Z^2
Consider (simple) random walk on $\mathbb{Z}^2$ started at the origin. The probability that the walk returns to the origin before hitting $(0,1)$ is $1/2$.
To see this, let $a(x)$ be the potential ...
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Marginals and Convex Sets
I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.
I have a collection of affine ...
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Bounding mutual information given bounds on pointwise mutual information
Suppose I have two sets $X$ and $Y$ and a joint probability distribution over these sets $p(x,y)$. Let $p(x)$ and $p(y)$ denote the marginal distributions over $X$ and $Y$ respectively.
The mutual ...
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Compact sets of the complex plane having the K-property ?
I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples.
As it is ...
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Standard way of determining if you have enough data to reliably compute success probability
Given $s$ successes in $n$ trials, where $p=\frac{s}{n}$, is there a standard way to determine if I have enough data to compute a meaningful statistic? For example, given $s=1, n=10, p=0.1$, the 95% ...
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Heavy Tailed Network
In his paper Kronecker Graphs: An approach to modeling Networks Jure et Al, mention that an important property of networks are that they are heavy tailed.
I'm trying to get an insight on what this ...
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1
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Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance
I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...
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BM and interpretation of stopping time sigma algebra
Suppose $H$ and $K$ are open subsets of $\mathbb{R}^d$ containing the origin with $H\subset K$, $B_t$ a standard Brownian motion starting at the origin, $\mathcal{F}_t$ its canonical filtration, and $\...
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What is the orthonormal basis for the Bergman space on the disk?
[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, ...
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Probability and information. The burrel-bucket-glass problem.
Suppose we have a barrel with three different kinds of marbles: red, green and blue. The probability to find a red marble in the barrel is R0, analogously the probability for green is G0 and for blue ...
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636
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Does anyone know an example of non-separable $L^1$ of a probability space?
It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
...
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The expectation of $\sqrt{B(n,p)}$
Let $n$ be a growing integer parameter, and suppose that $X_1,\dotsc,X_n$ are independent Bernoulli random variables with the probabilities of success $p_i:={\mathsf P}(X_i=1)$. If $X=X_1+\dotsb+X_n$ ...
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The finite-dimensional distributions of infinite-dimensional limit of finite-dimensional vectors
Suppose we have the process $\{\varepsilon_t,t\in \mathbb{N}\}$. Suppose that this the finite-dimensional distributions of this process are Gaussian, i.e. for any $t_1,...,t_n$, vector $(\varepsilon_{...
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Cover time for a biased random walk on an 'N'-dimensional integer lattice
Imagine that I have a random walk on an $N$-dimensional integer lattice, $Z^N$, of finite dimensions, $(d_1, ..., d_N)$, where boundaries are fully reflecting and the walker is initialized at some ...
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# bridges in random connected graph
Suppose we have an Erdos random graph with $n$ vertices and $c n$ edges.
What can you say about the probability that the graph is connected?
(More importantly) If it is connected, what is the ...
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Probability formula derivation
How is the following formula derived which yields the probability that the sum of the squares of n random draws from the closed interval [-1,1] is less than one?
formula: (1/2^n)*pi^(n/2)/(n/2)!
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Product rules are local and covariance identities are global
Start with the simple identity:
$$(f(x) - a)(g(x) - b) + a(g(x) - b) + b(f(x) - a) = f(x)g(x) - ab.$$
If $a$ and $b$ are the respective values of $f$ and $g$ at some point, then, after dividing both ...
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Is there a probabilistic interpretation of Dedekind zeta functions?
Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.
In the ...
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How can the topological entropy and $L^2$ mixing rate be related?
For a product of otherwise identical systems evolving at different rates, the toplogical entropy and a quantity very closely related to (indeed, identifiable with a nondegenerate variant of) the $L^2$ ...
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exactly simulating a random walk from infinity
In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is ...
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Extending state space to make a process Feller
Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...
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Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
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"Reverse" stochastic dominance
Let $\mu$ and $\mu'$ be probability measures on $\lbrace0,1\rbrace^\Lambda,\:\: \Lambda:= {\lbrace 0,1,\ldots,n\rbrace}$. Assume that
$\mu(X_i=1|X = \zeta \text{ on } \Lambda \setminus \lbrace i\...
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How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
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tight bounds on probability of sum of laplace random variables.
Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
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Is there a good explanation for this fact on pairwise independent variables?
(related question : most general way to generate pairwise independent random variables?)
Let $X_1,X_2,X_3,X_4$ be four random variables with standard Bernoulli distribution (i.e. $P(X_i=0)=P(X_i=1)=\...
- 3,595
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Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
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Covariance sign
Hi,
Is it true that $Cov[f(X),g(X)]>0$ where $X$ is a random variable of unbounded support and $f,g$ are two strictly increasing real functions? I think by Chebyschev integral inequality I must ...
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Counting subtrees of a random tree ("random Catalan numbers")
Given a rooted tree $T$ and an integer $k \geq 1$, let $N_k(T)$ be the number
of subtrees of $T$ containing the root and having exactly $k$ nodes (take $N_k(T)=0$ if $T$ has less than $k$ nodes).
...
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Mean of an experiment
Suppose we have a bag of n different balls, and each time m (m<n) balls are taken out for checking from the bag and put back. ...
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Probability of having a "perfect" game of Set
The card game Set has very simple rules (see here for rules), and it has prompted mathematicians to ask several questions. I will describe one of these questions. When the game ends, there are usually ...
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Sufficient conditions for independence based on moments
Let $P$ be the joint distribution of two random variables $X$ and $Y$, that both have support on $(0,1)$ (I am also interested in the case where $X$ takes values on $k$-dimensional simplex, but I ...
- 233
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Conditional distribution of the modulus of the output of AWGN channel given the modulus of the input
Hi everyone,
I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.
Assume that we have we have two ...
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A question on independence
For each natural number $n \geq 2$, define the set $A_n$ to be the set of points $p/n$ with $0 < p < n, \gcd(p,n) = 1$. Now define a sequence of independent random variables $X_1, X_2, \cdots$, ...
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715
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Minimum Mean Square Error (MMSE) and Mutual Information (I)
Consider this setting:
$Y=X+N$
where $N$ is a Gaussian standard random variable and $X$ is another arbitrarily distributed r.v. You can think of this $X$ as a message being transmitted over an AWGN ...
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