# 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.

5,500
questions

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71 views

### Weak convergence of conditional probabilities

Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the ...

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97 views

### Comparing mixing time of lazy and non-lazy Markov chains

Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That ...

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139 views

### Bounds on the mills ratio

How do I show the following bounds on the mills ratio :
$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \ $ for $ \ \ \ x>0$ ...

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229 views

### Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$

Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...

**6**

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**0**answers

77 views

### Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...

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146 views

### Lower bound Renyi divergence between two discrete probability distributions

I am trying to understand the proof of Lemma 1 in this paper (Section 9.2).
The proof shows that given a discrete probability distribution $P=(p_1,p_2,...,p_k)$ where $p_1 \geq p_2 \geq ... \geq p_k$,...

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57 views

### Different type of measurability of transition kernel

Let $(E,d)$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{P}(E)$ be the space of all probability measures on $(E,\mathcal{E})$. We eqiup this space with the ...

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49 views

### Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$

Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...

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36 views

### Probability that maximal elements has the same position in samples from correlated random variables

Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ ...

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84 views

### Kolmogoroff condition for truncated random variables

Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...

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117 views

### Quantitative CLT bound

Consider an independent collection of random variables $W_i, i=1,\dots,n.$ and let $Z \sim N(0,1)$. Roughly speaking, we know that $W_i$ are close in distribution to $Z$, say each is itself a sum of $...

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518 views

### Does a random sequence of vectors span a Hilbert space?

Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...

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76 views

### Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion

I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...

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80 views

### Comparison of Rademacher processes

Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$.
Let $\epsilon_1,\epsilon_2,\dots,$ be a Rademacher ...

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42 views

### Derivative of stochastic process in $L^p$ coincides with sample path derivative

In the article Random ordinary differential equations, by J.L. Strand (1970) (it is available at https://core.ac.uk/download/pdf/82447522.pdf), it is stated the following result, which relates ...

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42 views

### Sanov-type finite-sample bound on $KL(P\|\hat{P}_n)$

Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...

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76 views

### If a strong Markov process reaches a Borel set a.s., can it be restarted from that set?

Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$,
$$
P_x(\exists t\ge0 \text{ such that } X_t\in B)=1.
$$
My question: Does there exist a stopping ...

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66 views

### Independent identical distribution sequence

given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...

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87 views

### Bounds on difference between “logsumexp” and variance?

Let $Z$ be a random variable with finite moment-generating function $M_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define
$C_Z^\delta := \inf_{\...

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228 views

### Quadratic covariation of two not independent Brownian motions

Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...

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90 views

### Does the law of a Feller process depend continuously on the initial condition?

Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing ...

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88 views

### Two mixing rates of random dynamical system

Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...

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134 views

### Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
Furthermore, let
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$
be a $...

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64 views

### Smallest singular value distribution

Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...

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119 views

### Optimisation under constraint of Wasserstein distance

Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...

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69 views

### The weak version of the memoriless property

In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property
$$
P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right],
$$
...

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148 views

### Lower Bound of KL-Divergence Between Two Gibbs Measures

Suppose we have two Gibbs measures with densities
$$
p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...

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44 views

### Regularity of the pdf of partial Birkhoff sums

Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...

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55 views

### If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...

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160 views

### On Riemann integration of stochastic processes of order $p$

Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...

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102 views

### Density near at $0$ for the integral of the positive part of the Brownian motion

This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...

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120 views

### Difference of two probability measures modulo a third

Given three probability measures on $N$ elements (so $\mu_0, \mu_1,\mu_2 \in \ell^1_N$), I need to define the difference of $\mu_1$ and $\mu_2$ "modulo" $\mu_0$ as
$$
\sup \bigg\{ \int f \,\mathrm{d}(...

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65 views

### Strong convexity of internal energy with respect to Wasserstein metric

It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the ...

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74 views

### From quenched to annealed decay of correlation

If we have quenched decay of correlation, can we transfer it to annealed decay of correlation? To be precise, let us consider following setting:
Given transformations $T_{\omega}: (S^1, dm) \to (S^1, ...

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75 views

### A question about pdfs with likelihood ratio order

Suppose $f_1,f_2,\dots$ are pdfs of absolutely continuous random variables with the same support (say an interval). Assume that $\{f_i\}$ are strictly positive in their support. Furthermore, $\frac{...

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39 views

### Product of square-root of Wishart with an independent Gaussian matrix

We have two independent Gaussian random matrices with i.i.d. standard Gaussian entries each: $Q$ of size $k\times r$ and $P$ of size $\rho \times k$.
I'm interested in the following matrix: $(QQ)^{1/...

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103 views

### Is there a tight lower bound for the expectation of the product of two positive valued random variables?

Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...

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58 views

### Smoothness of expectation

Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...

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76 views

### boundig variation from median [closed]

Given a scalar random variable $X$, suppose that there are positive constants $c_{1}$ and $c_{2}$ such that
$$\forall t\geq 0 : \,\,\,\,\,\,\ \mathbb P\{|X-\mathbb EX|\geq t\}\leq c_{1}e^{-c_{2}t^{2}}...

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129 views

### Find probability of non-stationary inputs into Turing machine?

Consider some finite string $x=(x_1,x_2,...,x_{n-1},x_n)$ that is drawn from a non-stationary process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as,
$$
P_M(...

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134 views

### L1 distance after Convolution

Given two discrete distributions $P$ and $Q$ with the same support $x_1,\cdots,x_n$. Assume $K \in L^1(\mathbb{R})$ is a nonnegative function with $\int_\mathbb{R} K(x)dx = 1$, and let $K_h(x) = \frac{...

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66 views

### Large Deviations for Self-Normalized Sums

I am trying to understand the main result (Theorem 1.1) in this paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables
$$
\frac{\sum X_i}{\sqrt{n}\sqrt{\sum ...

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54 views

### Continuity of a continuous time martingale

Consider a Brownian motion $(B_t)_{t\ge 0}$ and its natural filtration $\sigma(B_t)$. Suppose $y_t$ is a $[0,1]$ valued, $\sigma(B_t)$ martingale. Does $y_t$ have continuous sample paths? If not, I ...

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57 views

### What is the Bruss-Yor concept of no information?

A few years ago, a question related to a paper of Thomas Bruss and Marc Yor on the so-called last arrival problem received some attention on this forum.
What I'd like to know now is:
What are the ...

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383 views

### What numbers can simulate 1/2?

Given two numbers $p,q\in(0,1)$, we say that $p$ can simulate $q$ if, given a biased coin with probability $p$, we can toss it a bounded number of times and use the results to simuate a biased coin ...

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125 views

### Local behaviour of fractions with bounded denominator / Was it already studied?

My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions ...

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103 views

### Knights on an n x n chessboard

In a n x n chessboard a white knight sits on the top left corner, and a black knight on the bottom right corner. Starting with white, the two knights take turns to move at random, and with equal ...

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377 views

### What does convergence in distribution “in the Gromov–Hausdorff” sense mean?

I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$
"...

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**1**answer

57 views

### What is the distribution of a Cartesian power of a collection of iid uniform points? (renewed)

The following question was asked recently at https://mathoverflow.net/questions/326631/what-is-the-distribution-of-a-cartesian-power-of-a-collection-of-iid-uniform-poi :
Take a rectangle with ...

**54**

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**2**answers

2k views

### Guessing each other's coins

I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...