# 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.

6,449
questions

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66 views

### Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...

**3**

votes

**1**answer

71 views

### Characterizing 'very homogeneous' finitely valued stochastic processes

Fix a positive integer $n$. Let $X = \{X_i\}_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint ...

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**1**answer

48 views

### Characterization of random variables whose tensor powers have subexponential “small-ball” probabilities

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
1. Symmetry: $\zeta \overset{d}{=} - \zeta$.
2. Small-ball probability: there exists ...

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**3**answers

455 views

### Taking points uniformly inside a general finite geometric domain

It is well known that if we want to take $n$ uniformly and randomly points inside a circle of radius $r$ and centered at the origin the following apparently correct approach for generating $x$ and $...

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**2**answers

527 views

### Recursive random number generator based on irrational numbers

Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $...

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123 views

### Why are financial markets modeled by càdlàg processes?

When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...

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139 views

### Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$

Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...

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**1**answer

55 views

### Diagonal terms in the Kochen Stone inequality

In a paper in Lecture Notes in Mathematics vol. 1874, Yan states the Kochen-Stone theorem in the following form, where $A_n$ is a sequence of events such that $\sum_{n=1}^\infty P(A_n) = \infty$:
$$
...

**5**

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**1**answer

168 views

### When is a function on symmetric positive definite matrices an expectation of Gaussian?

Is there some characterization of real-valued functions of the form $\Phi(C)=\mathbb{E}F(X)$, where $X$ has the Gaussian $N(0,C)$ distribution, on the space of symmetric positive semidefinite $n\times ...

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votes

**1**answer

82 views

### Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...

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**1**answer

63 views

### Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...

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**1**answer

68 views

### Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...

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**1**answer

52 views

### A scaled random walk on the number line

An agent $A$ is performing a random walk on the number line. Let $X_t$ be his position at time $t$. $X_{t+1}$ is calculated according to the following rules:-
$ X_{t+1} =$
\begin{cases}
...

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**0**answers

54 views

### Ergodicity properties of a linear operator on the Wiener Space

Suppose I have a centered Gaussian measure $\mu$ on a separable Banach space $B$. Let $H \subset B$ be its Cameron-Martin space and assume that it is dense in $B$.
Suppose that $U:H \to H$ is a linear ...

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158 views

### Does additive Gaussian noise preserves the Shannon entropy ordering?

Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy.
Does relation $h(X+Z) \geq h(Y+...

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**1**answer

90 views

### Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that
\begin{align}
E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...

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39 views

### Anti-concentration of measure: Slud's inequality for finite populations

Suppose that I have Bernoulli trials with unknown bias $p$. I need $\Omega(\frac{\log 1/\delta}{\epsilon^2})$ samples for the average of the samples to estimate $p$ within $\epsilon$ error with ...

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38 views

### A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...

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213 views

### How to compare pathwise convergence and convergence in probability

This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here.
Motivation: It appears pathwise convergence can ...

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**1**answer

109 views

### Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube

The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...

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**4**answers

553 views

### Self-contained formalization of random variables?

I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...

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48 views

### Reference request for invariance principles

In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...

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48 views

### Reference request: convergence of cadlag stochastic processes at $t=\infty$

Let $D\equiv D([0,\infty))$ be the space of cadlag functions (right continuous with left limits) on $[0,\infty)$. Consider a sequence of stochastic processes $\big(X^n\equiv (X^n(t))_{t\ge 0}\big)_{n\...

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65 views

### The reason why a test is undersized?

Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that:
$$n T_n \rightarrow_d \chi_K^2$$
under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...

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**0**answers

79 views

### Minimal perturbation of a Wigner matrix needed to produce an orthogonal top eigenvector

The instructor proposed a the following statement in the passing and suggested that we think about it (although it is not required):
For any $N \times N$ Wigner matrix, we replace $k$ entries with ...

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69 views

### Probability of satisfying the congruent mod equation

I'm wondering about the probability of picking three different numbers $x,y,z$ out of the set $[50]=\left\{ 1,2,3,...,50\right\}$ satisfying the equation: $$xyz\equiv \gcd(x,y,z)\mod 7$$ I started out ...

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61 views

### When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?

Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...

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**1**answer

58 views

### Is the set of almost surely continuous points dense?

Denote by $D(0,T)$ the space of right continuous functions with left limits defined on $[0,T]$. Let $\mathbb P$ be a probability measure on $D(0,T)$. Define
$$cont(\mathbb P):=\Big\{t\in [0,T]:~ \...

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116 views

### Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme)

In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$
$$R_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1_{x\geq y},$$
which is the Poisson distribution pdf. (This is ...

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91 views

### Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...

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33 views

### Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...

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44 views

### Concentration inequalities for gradient flows induced by random fields

Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...

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**1**answer

72 views

### GKS inequality with boundary condition

I want to know whether the following version of GKS inequality with boundary condition for Ising model hold or not. Consider Ising model on $\mathbb{Z}^d$ and $\varnothing \neq A\subset \Lambda_1 \...

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132 views

### Request for books/articles on random polynomials

Can somebody kindly recommend me a couple of introductory books/articles on random polynomials with clear expositions of fundamental results (like the distribution of roots, expected number of real ...

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**1**answer

89 views

### Ideas on how to prove Pythagorean identity involving Wasserstein distances?

I conjectured earlier that if $P$ and $Q$ were two probability measures, then we could show
$$W^2(P,Q) = \min_{T} [d^2(P,T_{\#}P) + W^2(T_{\#}P,Q)]$$ where $W^2(P,Q)$ denotes the squared Wasserstein-2 ...

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32 views

### How to use the mixed normal distribution to construct a proper statistics?

For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct
\begin{equation*}
\Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n
\end{equation*}
for ...

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48 views

### Projection onto column space perturbed by Gaussian noise

Suppose we have a matrix $X\in\mathbb{R}^{m\times n}$ (with $n \le m$) with iid standard Gaussian entries, and suppose we have noise matrix $W\in\mathbb{R}^{m\times n}$ with iid Gaussian entries, but ...

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59 views

### Poincare inequality for martingales

This is a vague question but here we go: Is there a form of the Poincaré inequality that is better suited for martingales?
For example, the Poincaré inequality for the boolean cube states that for any ...

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**1**answer

221 views

### Extension of Bernstein’s Inequality when the random variable is bounded with large probability

Bernstein’s Inequality can be stated as follows : Let $x_1, x_2, \dots, x_n$ be independent bounded random variables such that $\mathbb{E}[x_i] = 0$ and $|x_i| \leq \zeta$ with probability $1$ and let ...

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41 views

### Levy process approximation

Assume we have i.i.d. random sequence:$\{\xi_i\}$, denotes $S_n:=\sum_{i \le n} \xi_i$.
If $\exists m_n \to \infty$, a Levy process $(X_t)_{0\le t \le 1}$ and an independent unit-rate Possion process $...

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**1**answer

68 views

### If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$

Let $m$ be a large positive integer and $X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X_j$'s. For example, ...

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97 views

### Average over spheres finite

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$
I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...

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44 views

### A notion of (weak) dependency

Let $\nu$ be a probability measure over some product metric space $\mathcal{X}^n$
such that there exist some $\alpha \geq 0$ for which
for every $1\leq i \leq n$ it holds that
$\sup_{x\in \mathcal{X}^...

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**1**answer

79 views

### Convolution of two Gaussian mixture model

Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...

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142 views

### My hypothesis about convergence of series of independent random variable I cannot prove/disprove

Let $Y_i$, $X_i$ be sequences of independent random variables. Assume both limits exist: $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} \operatorname{Var}X_i}{\sum_{i=1}^{n} \operatorname{Var}Y_i},\quad \...

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45 views

### Label a random variable using an i.i.d

This question is also bountied on math-stackexchange.
I'm reading the paper Kesten’s theorem for Invariant Random Subgroups
and am trying to understand what exactly is meant by labeling the vertices ...

**2**

votes

**1**answer

76 views

### Diagonalizability of Gaussian random matrices

Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...

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53 views

### Uniqueness for measure valued ode

Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(N)$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's say :
$$ d\...

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**1**answer

84 views

### How to compute the following probability involving 4 normal random variables?

$\alpha, \alpha', \beta$ and $\beta'$ are four independent standard normal random variables, I am wondering how to compute the probability of the following two events:
$\alpha>\alpha'>0, \ \ \...

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47 views

### Sequence of random variables: relation between convergence and joint distributions

Let $X, X_1, X_2, \ldots $ be a sequence of $\mathbb{R}^d$-valued random variables defined on a common probability space $(\Omega, \mathscr{F}, \mathbb{P})$ such that the pairs
$$\tag{1}(X_n,X) \quad \...