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
7
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Random Functions and Transition Probabilities
Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability ...
7
votes
3
answers
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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
$$...
- 1,399
2
votes
2
answers
1k
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Independence of Brownian motion at hitting time from that hitting time
Let $B_t$ be a Brownian motion for a given probability space and $T:=\inf \lbrace t\geq 0 : \vert B_t \vert = 1 \rbrace$.
Is the process at this time, $B_T$, independent of the hitting time $T$? If ...
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4
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0
answers
331
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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 ...
- 445
0
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0
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493
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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 ...
- 667
3
votes
2
answers
513
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Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
- 201
12
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3
answers
1k
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Correlations in last-passage percolation
Consider the last passage percolation model on $\mathbb{Z}^2$ with, say, geometric weights on each edge. By a landmark result of Johansson (http://arxiv.org/abs/math/9903134), we know that if $T_n(\...
3
votes
0
answers
364
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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$$
Such that $m_1 \lt m_2 \lt \cdots \lt m_k$.
How many ...
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3
votes
2
answers
251
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Designing a tree to match a distribution
I want to design a tree to approximate a given
sequence of numbers, in the following sense.
Let $X=(x_1,\ldots,x_n)$ be $n$ numbers, with $0 < x_i \le 1$
and $\sum_i x_1 = 1$.
For a rooted tree $T$,...
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51
votes
3
answers
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What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
- 19.2k
1
vote
1
answer
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Structure of Measurable Subsets of the Unit Square
If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real ...
- 99
3
votes
0
answers
269
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Conditioning on the current value of a stochastic process
I want to condition a stochastic process $X$ on the current value of another process $Y$ in a continuous-time setting. So I am looking for a process $Z$ such that for every fixed $t$,
$$Z_t = E\big(...
- 103
1
vote
1
answer
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Convergence of Eigenvalues
Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest ...
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16
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5
answers
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Simple random walk on a locally finite graph: when is it recurrent?
I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...
- 30.3k
9
votes
1
answer
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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/...
3
votes
1
answer
4k
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Derivatives of conditional expectations
Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation $E_Y(a,...
- 8,512
5
votes
1
answer
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When is the limit of Martingales a Martingale?
I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$. Suppose that there is a filtration $F_t$ such that for each $n$, $X_n$ is a martingale with respect to this ...
- 195
3
votes
3
answers
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Stopping time of a Markov chain
Let $A(t+1)=A(t)+Bin(n-A(t),\frac{A(t)}{n})$ with $A(0)=1$ and let $T_n$ be the minimum of $t$ such that $A(t)=n$.
I think that $A(t)$ should behave like the naive deterministic approximation $a(t+1)=...
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3
votes
0
answers
107
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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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13
votes
2
answers
383
views
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.
...
- 7,857
2
votes
1
answer
319
views
Minimum distance to a sampled point with given pdf
Let $f(x)>0$ be a probability density function defined on the unit square $[0,1]^2$ in $\mathbb{R}^2$. Suppose that we take $N$ independent samples, $X_1,\dots,X_N$, of $f$. Now, sample a point $...
- 645
6
votes
2
answers
745
views
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 ...
- 7,527
1
vote
2
answers
582
views
White Noise Space and Local Time
This question follows from the answer I gave to the question "Wiener Meets Sobolev" in the MathStackExchange Forum.
I was wondering in the context of White Noise Space if the Local Time at x of a ...
- 1,334
1
vote
1
answer
284
views
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
$S_n:=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i.$
...
- 1,491
1
vote
0
answers
620
views
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 ...
- 27
3
votes
2
answers
462
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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 ...
- 371
1
vote
0
answers
236
views
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?
- 922
2
votes
3
answers
292
views
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 ...
1
vote
2
answers
170
views
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
...
- 3,475
7
votes
3
answers
995
views
Kolmogorov probability axioms without non-negativity condition
What is a minimal consistent modification of probability axioms to include negative values?
Is it enough to use a minimal modification of axioms obtained by
formal exclusion of non-negativity ...
- 211
26
votes
4
answers
2k
views
A percolation problem
Let's consider the 2-dimensional integer lattice $\mathbb{Z}^2$ for simplicity. In "ordinary" bond percolation, there is a parameter $p \in [0,1]$, and each edge is on with probability $p$. Consider ...
- 983
24
votes
1
answer
615
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Permutations, stopping times, Bessel functions, hook formula and Robinson-Schensted
For given counting number $n$, consider all permutations $\pi$ of {$1,\ldots,n$}, generate for every $\pi$ its Robinson-Schensted pair of standard tableaux $(P_\pi,Q_\pi)$ and average together all the ...
- 17.6k
11
votes
3
answers
666
views
Limit shape for fixed-perimeter lattice polygons
Let $P$ be a simple polygon defined by $n$ unit-length segments
connecting lattice points of $\mathbb{Z}^2$.
I have two operations that preserve the perimeter of $P$.
The first is the "pop" of a ...
- 151k
2
votes
1
answer
162
views
Question on two measures of correlation
For two $\sigma$-fields, $\mathcal{A}$ and $\mathcal{B},$ we have the notion of HGR maximal correlation
$$\rho(\mathcal{A},\mathcal{B}) = \sup \frac{Efg-Ef.Eg}{\sqrt{\mathsf{Var}(f).\mathsf{Var}(g)}}...
- 21
8
votes
3
answers
749
views
non-integrable subadditive ergodic theorem
Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...
- 23.2k
4
votes
1
answer
922
views
Convergence in probability only depends on topology?
Suppose $(S,d)$ is a Polish space, and $X$, $(X_n)$ are random variables such that $X_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $...
- 2,895
3
votes
0
answers
206
views
representing vine copulas
Vine copulas is a way to represent multidimensional distributions (n-densitys)
as a product of the n 1-marginal densities and a product of (n choose 2) bivariate copulas, where som of them are ...
- 2,600
2
votes
0
answers
979
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How to calculate/approximate expectation of function of a binomial random variable?
Hi,
I am stuck at following problem in my research.
Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\...
- 21
1
vote
1
answer
3k
views
expected value of inner products of iid standard normal vectors
Hello,
I wish to calculate (or upper bound) expectations of the form $E[\langle x,y \rangle^2]$, where $x$ and $y$ are i.i.d standard gaussian vectors of length n. Are there any exponential type ...
- 119
2
votes
1
answer
290
views
Negatively associated point processes
A point process $\Phi$ is said to be negatively associated if for any finitely many bounded Borel subsets $B_1,B_2,...,B_n,$ we have that
$$\operatorname{Cov} \left( f\left(\Phi\left(B_1\right),\...
- 113
3
votes
0
answers
350
views
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 ...
- 667
4
votes
3
answers
2k
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Probability of overlapping of repetitive events
The question is to compute or estimate the following probabilty.
Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If ...
- 141
0
votes
1
answer
285
views
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 $X$ ...
- 13
1
vote
1
answer
302
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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+\...
- 11
5
votes
1
answer
839
views
An optimization problem involving sum of binomial coefficients upto some value
I would like to minimize $f(s, n, \epsilon)$ with respect to $s$ where
$$f(s,n,\epsilon) = \left( 1 + \frac{n}{2^s} \right)\frac{1}{s} \sum_{k=0}^{\lfloor s\epsilon \rfloor} {s \choose k}~.$$
Note ...
- 362
1
vote
1
answer
559
views
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)...
- 51
1
vote
2
answers
290
views
Getting $B_t$ from its local times $L^x_t$
Hi
Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?
Using occupation time formula this would mean solving for some $f$ the following ...
- 1,334
3
votes
3
answers
2k
views
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 $E(X) = \frac 1p$, $\...
- 8,903
2
votes
2
answers
655
views
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 ...
- 2,221
6
votes
3
answers
814
views
A simple stopping time problem.
This should be rather standard so I hope somebody with a good background in probability theory would give me a quick solution or a reference.
We are given a threshold positive integer $T>0$. Let $...
- 195