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.
9,025 questions
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Is this probability distribution known in the literature?
In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution?
$\Pr(X\le x)=\exp\left[-\frac{1}{2}\left(\frac{1}{2}x-\sqrt{1+\frac{x^{2}}{4}}\right)...
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Distribution of big component of set partitions
Consider the set $S_n = \{1, \dotsc, n\},$ and consider the set $P(n, k)$ of partitions of $S_n$ into $k$ parts (the cardinality of $P(n, k)$ is the Stirling number of the second kind $S(n, k).$ ...
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Maximum vertical distance for a lattice path when NSEW steps are allowed
Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate ...
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Interesting applications of [Martingale/Brown motion/diffusion/percolation ] theory?
This question is motivated by an exercise called "The Star-ship Enterprise's Problem" in Williams's book "probability with martingales", it can be stated as follows:
Suppose the control system on ...
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Help prove a maximal inequality
Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number.
$S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$.
I want to prove:
$$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) \...
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
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Positive martingale representation with jumps
I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question:
Martingale representation theorem for Levy processes
My best guess is (subject to ...
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Selecting two random points inside a sphere which are a fixed distance apart
Without appealing to a guess-and-check approach, how might I select a pair of random points inside of a sphere of radius $R$ s.t. the points always a distance $d \leq R$ apart? Can the selected ...
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Density of countably additive measure in the set of all finitely additive measures.
Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
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Integrated colored Gaussian noise
Assume we have a colored Gaussian process $z_t$, with an autocorrelation function $cov(z_t,z_s)$ given by an analytical function $\alpha(t,s)$ (if it helps, one can assume that $\alpha(t,s) = \kappa e^...
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computing average height-functions for lozenge tilings
Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to ...
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bounding the probability that a polynomial is near 0
Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
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A non-degenerate martingale
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.
Let $Y_t$ be a martingale given by
$$...
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Distribution of the sizes of conjugacy classes in the symmetric group.
This recent question makes me wonder: is there some known limit theorem for the distribution of the sizes of conjugacy classes in the symmetric group $S_n?$ A quick search seems to reveal nothing ...
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Sum of a Gaussian and an independent second moment constrained random variable
I am studying the (asymptotic) behavior of the p.d.f of the random variable $Y = X + Z$, where $X$ is an r.v. with any distribution function $F(x)$ such that $\int_{-\infty}^{\infty} x^2 \mathrm{d}F(x)...
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What is 'arch' in Vershik-Kerov's 1984 paper?
In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperator{\arch}{arch}\arch ...
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Simulating conditional expectations
There is a multidimensional process X defined via its SDE (we can assume that its a diffusion type process), and lets define another process by $g_t = E[G(X_T)|X_t]$ for $t\leq T$.
I would like to ...
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Coin flipping and a recurrence relation
How can one solve the following recurrence relation?
$f(n) = 1 + \frac{1}{2^n} \sum_{k = 0}^n {{n}\choose{k}} f(k)$
$f(0) = 0$
As it happens, I can show $f(n) = \Theta(\log n)$ through other means (...
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Log concavity of noncentral chi-square
I am trying to check if the noncentral chi-square distribution is log-concave in its noncentrality parameter. Specifically, given
$p(y ; \lambda, \sigma^2) = \frac{1}{2\sigma^2}\exp\left(-\frac{y+\...
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Best introduction to probability spaces, convergence, spectral analysis
I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP:
probability ...
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combination and probability
There are $k$ sets of numbers:
$$\lbrace0,1,2,\ldots,m_1\rbrace, \lbrace0,1,2,\ldots,m_2\rbrace, \ldots,\lbrace0,1,2,\ldots,m_k\rbrace$$
Such that $m_1 \lt m_2 \lt \cdots \lt m_k$.
How many ...
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Comparing two measures on trees on $n$ vertices
A standard measure on trees on $n$ vertices is the Uniform Spanning Tree (UST) on the complete graph. This is the measure where every tree has equal probability, $1 / n^{n-2}$ by Cayley's formula.
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Designing a tree to match a distribution
I want to design a tree to approximate a given
sequence of numbers, in the following sense.
Let $X=(x_1,\ldots,x_n)$ be $n$ numbers, with $0 < x_i \le 1$
and $\sum_i x_1 = 1$.
For a rooted tree $T$,...
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Conditioning on the current value of a stochastic process
I want to condition a stochastic process $X$ on the current value of another process $Y$ in a continuous-time setting. So I am looking for a process $Z$ such that for every fixed $t$,
$$Z_t = E\big(...
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Structure of Measurable Subsets of the Unit Square
If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real ...
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Pólya's Random Walk Constants at infinity
Let be the probability that a random walk on a d-D lattice returns to the origin. In 1921, Pólya proved that $p(1)=p(2)=1$
but $p(d)<1$ for $d>2$.
http://mathworld.wolfram.com/...
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Linear relations with small coefficients
NOTE: Slightly more general question follows my specific one at the top
For $1 \leq i,j \leq n$, let $e_{ij}$ be the vector in $\mathbb{R}^{n^{2}}$ with a 1 in entry $(n-1)i + j$, and 0's everywhere ...
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What is the probability for a random algebraic cycle to be homologically trivial?
Someone recently asked me about the Hodge conjecture. As I understand it the conjecture asserts the existence of many non trivial algebraic cycles. The difficulty comes from the fact that we don't ...
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Topple height of randomly stacked bricks
What is the expected height of a stack of unit-length bricks, each one
stacked on the previous with a uniformly random shift within $\pm \delta$?
The stack topples if the center of gravity of the top $...
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Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
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Multivariate CLT, convergence of densities
Let $X_i$ be a sequence of i.i.d. $\mathbb{R}^d$-valued, continuous (i.e. with density) random variables. We assume that $E X_i =0$ and $Cov(X_i)=Id$. Let
$S_n:=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i.$
...
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using distribution of primes to generate random bits?
In his popular science book The Music of the Primes, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair ...
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Motivating the use of the theory rough paths in stochastic analysis
I am a final year undergraduate looking to do a PhD in stochastic analysis, perhaps with applications to problems in mathematical finance. On a potential supervisor's webpage, it says that one of his ...
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density for Gaussian gram matrices
Let $Z \sim \mathcal{N}(0,\Sigma \otimes I)$ (so the columns of $Z$ are distributed $\mathcal{N}(0, \Sigma)$) and $A = Z'Z.$ Is there a name for the distribution on $A$? Is the density known?
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non-integrable subadditive ergodic theorem
Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...
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3
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2d moment of chebyshev
Given: $X_i$ are independent {-1,1} variables with expected value 0 and $X = \sum_{i=1}^n X_i$
Is there a closed form solution / tight bound for $E[ X^{2d} ]$ ?
I realize this problem sounds very ...
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Bound on expression from probability distributions
I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression
...
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Convergence in probability only depends on topology?
Suppose $(S,d)$ is a Polish space, and $X$, $(X_n)$ are random variables such that $X_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $...
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Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog
I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...
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Randomized algorithm?
The problem is as follows. Given a set $S$ of natural numbers of size $n$ where each $x_i \in S$ is from the set $[n^2]$. Elements of $S$ are not necessarily pairwise different, i.e., there can be ...
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representing vine copulas
Vine copulas is a way to represent multidimensional distributions (n-densitys)
as a product of the n 1-marginal densities and a product of (n choose 2) bivariate copulas, where som of them are ...
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How to calculate/approximate expectation of function of a binomial random variable?
Hi,
I am stuck at following problem in my research.
Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\...
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expected value of inner products of iid standard normal vectors
Hello,
I wish to calculate (or upper bound) expectations of the form $E[\langle x,y \rangle^2]$, where $x$ and $y$ are i.i.d standard gaussian vectors of length n. Are there any exponential type ...
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Is there a monotone coupling of Dirichlet random variables?
Let $X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$ and $Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$, where all $a_i$ and $b_i$ are positive. Is there a natural coupling between $X$ ...
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Observing drift of a Levy process
It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy ...
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Conditional geometric distributions
If $p<1$ and $X$ is a random variable distributed according to the geometric distribution $P(X = k) = p (1-p)^{k-1}$ for all $k\in \mathbb{N}$, then it is easy to show that $E(X) = \frac 1p$, $\...
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Law of large numbers for stochastically chosen samples
Let $X_t$ be a sequence of i.i.d. random variables with mean $\mu$. Then the law of large numbers states that
$$\lim_{T \to \infty} \frac1T \sum_{t=1}^T X_t = \mu \quad a.s.$$
Now suppose that (in a ...
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Special case of Duffin-Schaeffer conjecture
The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: http://en.wikipedia.org/wiki/...
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Measure on $\omega_1$
Let $\mathcal{O}$ be the $\sigma$-algebra on $\omega_1$ generated by its countable subsets. Is there a ($\sigma$-additive) probability measure on $\mathcal{O}$ that is not concentrated on a countable ...
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Distribution of the standard deviation of normal variates
What is the distribution of the standard deviation of $n$ normal variates? That is, if $X_1,...,X_n$ are i.i.d. normal random variables with mean $\mu$ and s.d. $\sigma$ and $M=\sum X_i/n$, then what ...
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