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.
7,576
questions
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Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent [closed]
Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$.
Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...
- 93
-1
votes
2
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93
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Cumulants of a sequence of variables with zero mean and variance
Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
- 11
0
votes
1
answer
62
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Functional relationship between two quantities
Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by
$$
\alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
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6
votes
1
answer
409
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Max decoupling inequality
Let $X_1,\ldots,X_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X_1,\ldots,\tilde X_n$ be their independent version: $\mathbb{E}X_i=\mathbb{E}\tilde X_i$ for ...
- 5,101
3
votes
2
answers
125
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Continuity of Radon transform w.r.t the angle
Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by
$$
R[f](w,b) := ...
- 5,580
0
votes
0
answers
72
views
$L^p$ inequality for "positively correlated" random variables
Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
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3
votes
0
answers
85
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Does smoothing a non-log-concave distribution make it more log-concave?
Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded:
$$-\nabla^2 \ln p\succeq L$$
for some matrix ...
- 341
3
votes
0
answers
48
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Random assignment problem under multinomial or Poisson distribution
We place $m$ balls at random (uniformly) inside $n^2$ urns arranged as a $n \times n$ square. Then we must choose $n$ urns, such that no two urns belong to the same row or column, with the objective ...
- 298
1
vote
1
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39
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Estimation of Lévy measure of ID distribution
Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R_+$. Due to Lévy-Khintchine, its moment generating function then writes :
$$M(t) = \mathbb E\left(e^{tX}...
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0
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80
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Stopping times for martingale
The nonnegative integer set is denoted by $\mathbb{Z}_+$.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$ be an increasing sequence ...
- 51
1
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0
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91
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On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
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0
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43
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On the Markov property of a limit process
Let $E$ be a locally compact separable metric with countable base. We consider a sequence of Hunt processes $\{X^{(n)}\}_{n \in \mathbb{N}}$ on $E$. That is, each $X^{(n)}=(\{X_t^{(n)}\}_{t \in [0,\...
- 703
2
votes
1
answer
118
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Verify if array is orthogonal
This is a repost from the computer science stackexchange. The question has been offered a bounty, but received no answers. Therefore, I would like to ask this question here.
Orthogonal arrays often ...
- 470
0
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0
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22
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L^2 approximation error in Gaussian Process Regression (finite data setting)
I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
2
votes
0
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37
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techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$
We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...
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7
votes
2
answers
185
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On permanent of a square of a doubly stochastic matrix
Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
- 339
1
vote
0
answers
41
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Counting overlaps of random intervals
We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...
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1
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0
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122
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A basic formula for the falling factorial
Whis is a question I asked on Math.SE, but didn't get any response.
Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e.
$$
X(\omega): = \sum_{A \in \...
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0
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29
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How to use Itô's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\left<\mathbf{J}x_u, x_t\right>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?
I am reading a lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it shows that for $s\ge t$,
$$
K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}...
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3
votes
2
answers
118
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On finding an upper bound on the error of a sparse approximation
I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer.
Original question: https://math.stackexchange.com/...
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0
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0
answers
18
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Vector Norms of Sub-Gaussian Matrix Multiplication?
Let $X, Y$ be $n\times n$ matricies with i.i.d. sub-Gaussian entries.
I am interested in tail bounds for $\lVert XY\rVert$, where $\lVert\cdot\rVert$ is a norm on $\mathbb{R}^{n^2}$, i.e. I want tail ...
- 702
4
votes
1
answer
134
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Population growth with good and evil children - probability good outnumbers evil
Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those ...
- 245
1
vote
1
answer
76
views
Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
- 5,580
1
vote
0
answers
46
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Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 1
4
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1
answer
210
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How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?
I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging.
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...
- 156
0
votes
1
answer
128
views
Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,173
1
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0
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30
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Handling sums of correlated random variables with a directed path structure
Recently, I've been seeing random variables with the following correlation structure based on directed paths on a graph. For example, there are $16$ directed paths "directed downwards from the ...
- 245
1
vote
1
answer
54
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Non-independent Sub-gaussian variables and concentration
Let $g \in R^{d}$ have $iid$ Gaussian components. Let $a \in R^{d}$, and let $b \in R^{d}$. be arbitrary vectors.
Consider the random variable $Y_{g,g}:= \frac{1}{n}\langle g,a \rangle \langle g, b \...
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0
votes
1
answer
48
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Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?
Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
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0
answers
61
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Converse to Cameron-Martin theorem
It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....
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5
votes
1
answer
107
views
Second Skorokhod embedding in high dimensions
The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...
- 651
0
votes
0
answers
29
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Minimum eigenvalue of covariance matrix under probability constraint
Consider a random (column) vector $X\in \mathbb{R}^d$. I am interested in the quantity
$$
\Lambda(\alpha)=\inf_{E\in \mathcal{B}(\mathbb{R}^d), \ P(X\in E)\ge \alpha} \lambda_{\min}\left(E[XX'1\{X\in ...
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2
votes
1
answer
68
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Chung's law of the iterated logarithm for Brownian motion
I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...
- 121
0
votes
0
answers
45
views
A sufficient condition for the decomposition of a bounded random vector
Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{(x_1,x_2,\ldots,x_n)|\sum_{i=1}^m{\bf{a}}_ix_i,x_i \in [-1,1]\}$, where ${...
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A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 11
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0
answers
60
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Expectation of edge weights on the complete graph, Part 2
This question concerns the same basic set-up as my previous question: Expectation of edge weights on the complete graph
In that question an answer was given which shows that the expected value is as ...
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2
votes
0
answers
50
views
Approximate logarithmic bound on expected maximum via central limit theorem
If $Z_i$ are standard normal, possibly dependent, one can show that
$$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$
I'm looking for a similar (asymptotic) bound for asymptotically normal ...
- 213
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0
answers
39
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Total variation convergence of eigenvalues in the bulk of the GUE?
Let $\lambda_{k(n)}$, $k(n) / n \to a \in (0,1)$, be an eigenvalue in the bulk
of the $n \times n$ Gaussian Unitary Ensemble (GUE) normalized so that its spectral measure converges to the semi-circle ...
-1
votes
0
answers
178
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Open problems in derivatives, options and finance [closed]
I would like to ask what kind of problems remain open in finance. In particular, those concerning pricing derivatives, computing the risk, etc. It could be really of great help if you can recommend to ...
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1
vote
1
answer
131
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Expectation of edge weights on the complete graph
Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
- 21.3k
2
votes
1
answer
87
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
- 162
4
votes
1
answer
70
views
Distance between trunctated random walk and its normal form
I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$
$$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...
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3
votes
3
answers
465
views
How close are two Gaussian random variables?
Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
- 159
1
vote
0
answers
68
views
On the closedness of a certain subset of $\mathbb R$
Let $\mu$ be a probability measure on measurable space $X=\mathbb R^n$ (euclidean), and let $F$ be a family of $\mu$-measurable functions $X \mapsto \mathbb R$ which are uniformly bounded, i.e $b:=\...
- 5,580
0
votes
0
answers
61
views
Extending proofs from Lebesgue measure to non-atomic (probability) measures [closed]
On $\mathbb R^n$, is there a relationship between non-atomic probability measures and the Lebesgue measures ?
What kinds of results about non-atomic measures have the same proof as for Lebesgue ...
- 5,580
2
votes
1
answer
62
views
Compactness of the integral of a set-valued function
Let $X$ be a compact space (e.g. a compact subset of $\mathbb R^n$) and $P$ be a probability measure on $X$. Let $A$ be a compact subset of some $\mathbb R^d$. Finally, let $F$ be the collection of $P$...
- 5,580
1
vote
1
answer
54
views
Independent Sums and Orlicz Norms
Let $X_{i}$ be a collection of iid random variables of cardinality $n$, and let $S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}.$
Let $|| X||:=\inf_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-...
- 117
3
votes
0
answers
33
views
Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient
Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...
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votes
0
answers
29
views
How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?
I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...
7
votes
2
answers
182
views
Does entropy of the random walk control the return probability
Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n_x(y)$ is the probability ...
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