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.
8,658
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Extension of subharmonic functions at infinity
Let $W$ be the complement of a compact set $K$ in $\mathbb{R}^{n}$, and $u$ a subharmonic function on $W$. Can we find, under some conditions, a function $\tilde{u}$ that is subharmonic on $W\cup\{\...
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Does $\sqrt{n}\sup_x |\Phi(\dfrac{x-\overline{X}}{S})-\Phi(x)|\to0$ in probability?
Let $X_1,...,X_n$ be iid observations from $N(0,1)$. Let $\overline{X}=\dfrac{1}{n}\sum_{i=1}^n X_i$ and $S^2=\dfrac{1}{n}\sum_{i=1}^n (X_i-\overline{X})^2$. Then is it true that $\sqrt{n}\sup_x |\Phi(...
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A question on the problem of Dirichlet 2
Let $U$ be an open set in $\mathbb{R}^{n}$ with $n\geq2$ and $V$ an open set containing the boundary $\partial U$ of $U$. Suppose $u$ is subharmonic on $V$. We know that the generalized solution of ...
- 411
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1
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Any way to prove Monotone Continuity?
This question is cross-posted on math.stackexchange.
Let $A \succeq B$ be defined as $A$ is at least as probable as $B$. $\succeq$ obeys the following axioms. For any $A, B, C$,
Non-negativity: $A \...
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Does convergence in law imply convergence in convex distance?
For two random variables $X$ and $Y$ taking values in $\mathbb{R}^m$, the convex distance $d_c$ is defined as
$$d_c(X,Y) = \sup_{h} \lvert \operatorname{E}(h(X)) - \operatorname{E}(h(Y)) \rvert,$$
...
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Gaussian concentration inequality
Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in this paper. Specifically, Lemma 4 on page 307 states (without a proof) that
There exists a universal constant $...
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365
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Matrix positive semi-definite
We construct a non-random matrix using random variables as follows:
We fix the vector $v=(1,1).$
Let $X$ be a $\mathbb R^2$-valued random variable such that $X$ is distributed according to
$$d\mu(...
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0
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252
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General Hoeffding inequality with two uncorrelated random vectors
Let $X_1,\ldots, X_n$ be independent sub-Gaussian variables with sub-Gaussian norm $K$. Let $a_1,\ldots,a_n$ be a random vector such that $\mathbb{E}[a_j X_i]=0$ for all $i,j\in \{1,\ldots ,n\}$ and ...
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3
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When do $\phi^2$ and $\phi’^2$ have the same expectation under a Gaussian random variable?
I am looking for a function $\phi(x)$ such that
$\mathbb{E}_{x\sim\mathcal{N}(0,1)}[\phi(x)^2] = \mathbb{E}_{x\sim\mathcal{N}(0,1)}[\phi'(x)^2]$.
Obvious solutions are $\phi(x) = x$ and $\phi(x) = \...
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Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 506
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2
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Concentration of sum of concentrated random variables
I have a sum of positive random variables, they are not identically distributed, but even that case would be interesting. They are not necessarily independent, but I already have a concentration bound ...
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Strong data-processing inequality ? Upper bound on a certain modified total-variation metric
Let $\mathcal X=(\mathcal X,d)$ be a Polish space equipped with the Borel sigma-algebra. Let $p\ge 1$ and $P_1,P_2$ be probability distributions on $\mathcal X$ such that $\max_{k=1,2}\int d(x,x_0)^...
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Hanson-Wright inequality with random matrix
I'm interested in bounding the tail probabilities of a quadratic form
$x^t A x$ where $x\in \mathbb{R}^n$ is a sub-Gaussian vector with independent entries. $A\in \mathbb{R}^{n\times n}$ is a matrix. ...
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What is the largest possible probability that a random matrix over $\mathbb{F}_2$ is non-singular?
Suppose $A(p, n)=(a_{ij}(p))_{i, j \leq n}$ is an $n\times n$ random matrix over $\mathbb{F_2}$, with all its entries being i.i.d. and such that $P(a_{ij}(p) = 1) = p$, where $p$ is some real number ...
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Constant Martin kernel and amenability
Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
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Is the space of continuous functions from Polish space to Polish space Polish?
Theorem 4.19 in Kechris' Classical Descriptive Set Theory says that the space of continuous functions from a compact metric space to a Polish space is Polish. It is therefore obvious that the space of ...
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A determinantal mixture of probability densities
I came up with this operation after playing with determinantal point processes:
Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set
$$
f\star g(x)...
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$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
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2
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210
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Limited sum for whole sum approximation
Let $d_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to ...
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Law of large numbers for random Dirac measures
Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.
...
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Distance between value function of deterministic and stochastic control problems
Suppose that one wants to control a diffusion process
$$
dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x
$$
in order to optimize a stochastic control problem with value function
$$
V_T(u)=\...
- 4,969
3
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251
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Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,639
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487
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A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
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Is there a lower bound in the opposite direction of the data processing inequality?
Let $f_1$ and $f_2$ be the distributions of random variables $X_1$ and $X_2$ respectively. Let $g_1$ and $g_2$ be the distributions of $T(X_1)$ and $T(X_2)$ respectively, where $T(\cdot)$ is some ...
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Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 624
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1
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Continuum percolation in 1d
What is known about continuum percolation in 1d?
By this, I mean, for $d \in \mathbb{N}$, the Poisson-Boolean model of disks of radius $r_0 \in \mathbb{R}$ with centres arranged randomly in $[0,1]^{d}...
- 612
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1
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105
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Concentration of maxima of a random polynomial with Rademacher coefficients
Let $X_1,\ldots, X_n$ be independent Rademacher random variables (i.e. $\mathbb{P}(X_i=\pm 1)=1/2$). Consider the random polynomial $$P_{n}(t)=c+X_{1}t+X_2t^2+\cdots+X_{n}t^n.$$
Is it well known how ...
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Representation of optimal controls as diffusions
In reading this post I couldn't help but wonder the following question:
Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem:
$$
\begin{...
- 4,969
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A randomized central limit theorem
Let $X_k$, $k = 1, 2, \dots$, be a sequence of i.i.d. random variables with finite second moments. Also, let $N_k \geq 1$, $k = 1, 2, \dots$, be a sequence of random variables taking integral values, ...
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Random matrix properties
Let $\mathbf{H}_{N,K}$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $1$. Then, we know from the law of large number that if $N,K\rightarrow\infty$, we ...
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Distance between two sample quantiles
Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
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Small set in partition-large class
A collection $\mathcal{A}\subseteq \mathcal{P}(X)$
is $k$-large in $X$
if for every $k$-partition
of $X$ namely
$X_1,\cdots,X_k$, there exists an $i\leq k$ such that $X_i\in \mathcal{A}$;
$\mathcal{...
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Confidence Intervals for strongly mixing stochastic processes
I have $s(t)$, a stationary stochastic process that we know is strongly mixing - and I also know that samples from $s(t)$ are definitely correlated over time. I want to estimate the mean of $s(t)$, $\...
3
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1
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650
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Characteristic function and moments
Let $X\in L^1(\Omega)$ and $\phi_X$ the corresponding characteristic function.
We know that: $\phi_X$ is $n$ times differentiable (at $u=0$) iff $\mathbb{E}[X^n]<\infty$. (This depends a bit on ...
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Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution
$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...
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What is the minimal $m$ for which the independence graph is $n$-universal?
Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent.
...
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Less regular version of the Gaussian free field
One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
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Imprecise Definition of a $\sigma$-algebra
I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
- 3,151
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Reconstructing probability distribution with high probability
Sample $m$ times from unknown probability distribution $p=(p_1,p_2,\cdots,p_n)$, we can construct a probability distribution $q=(q_1.q_2,\cdots,q_n)$.
How large $m$ should be to achieve that the ...
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340
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Relate the solid angle and surface measure of a surface
Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...
- 161
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Gaussian Property of the Renormalization Group
Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, ...
- 3,151
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Properties of convolutions
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
and the function
$$h_p(x):=e^{-\vert x \vert^p}.$$
My goal is to analyze
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
- 173
2
votes
1
answer
175
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Non-convergence to a Gaussian
Let $f_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f_n(x,x)=0.$
I would like to know: Can we show that a ...
- 183
3
votes
2
answers
405
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Two independent function when considered as random variable over $([0,1],\mathrm{Lebesgue},\mathcal{B}_{[0,1]})$
Does there exists two non-constant continuous functions $f,g:[0,1]\rightarrow \mathbb{R}$ so that they are in independent (in probability sense) when viewed as random variables over the measure space $...
- 329
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1
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144
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Sphere inversion in Riesz potential
I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554).
On page 546, the authors ...
- 3
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0
answers
1k
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Approximate Multivariate Gaussian Integration by Parts
When $Z$ is a $\mathcal{N}(0,1)$ random variable, $f$ smooth from $\mathbb{R} \to \mathbb{R}$ we have the Gaussian integration by parts formula
$$
\mathbb{E}(Zf(Z)) = \mathbb{E} f'(Z).
$$
One analog ...
- 21
2
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0
answers
167
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Weak convergence of $\mathcal{L}^2$ valued random variables
Consider two continuous functions $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence ...
- 83
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1
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161
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Visualization of the disintegration theorem [closed]
Where can I find a picture that gives a visualization of the disintegration theorem?
If such reference does not exist, what would a nice visualization of this fundamental result look like?
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0
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612
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is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
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