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.

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Is independence preserved if a random entry in an independent sequence is replaced by a constant?

Let $\xi=(\xi_1,\ldots,\xi_n)$ be a sequence of independent random variables. Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi_\nu$ by a constant $c$. The rest of the $\xi_i$ ...
Andras Farago's user avatar
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1 answer
396 views

If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set

If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
0xbadf00d's user avatar
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Given a set of marginals, what is the largest support of a distribution satisfying these?

Given a random variable $X$ with support over $\{0,1\}^I$, we can define the marginal distribution on the bits indexed by $A \subseteq I$ by $Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup ...
Samuel Schlesinger's user avatar
1 vote
1 answer
3k views

Correlation between square of normal random variables

Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2 $ and $Y^2$ ? I am aware of the fact that the square of the normal ...
John's user avatar
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3 votes
2 answers
762 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 ...
JohnA's user avatar
  • 680
2 votes
1 answer
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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 ...
Josh R's user avatar
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2 answers
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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$ ...
John's user avatar
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5 votes
1 answer
257 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<...
Bedovlat's user avatar
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8 votes
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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 ...
Dmytro Taranovsky's user avatar
3 votes
2 answers
301 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 $p_1 \geq p_2 \geq ... \geq p_k$,...
Elwood Crandall's user avatar
4 votes
1 answer
120 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 ...
Oleg's user avatar
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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 ...
dohmatob's user avatar
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1 vote
1 answer
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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 $L$ ...
mkatkov's user avatar
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1 answer
226 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 ...
Maximilian Janisch's user avatar
3 votes
1 answer
194 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 $...
passerby51's user avatar
  • 1,639
8 votes
2 answers
618 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 ...
J. E. Pascoe's user avatar
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3 votes
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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 ...
E_Wijler's user avatar
5 votes
1 answer
165 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 ...
user58955's user avatar
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1 vote
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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 $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$. ...
dohmatob's user avatar
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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 ...
user1118's user avatar
1 vote
1 answer
72 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, \...
jason's user avatar
  • 553
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1 answer
289 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_{\...
dohmatob's user avatar
  • 6,716
0 votes
2 answers
5k 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 ...
Charlie Shuffler's user avatar
1 vote
0 answers
150 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 ...
Potato's user avatar
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1 vote
0 answers
172 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 ...
jason's user avatar
  • 553
2 votes
2 answers
235 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 $...
Joker123's user avatar
  • 153
2 votes
0 answers
95 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 ...
Dominik's user avatar
  • 83
5 votes
3 answers
464 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 ...
SiXUlm's user avatar
  • 101
1 vote
1 answer
89 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 $$ P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right], $$ ...
Matjaž Krnc's user avatar
8 votes
1 answer
311 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 $g$, ...
Minkov's user avatar
  • 1,117
1 vote
0 answers
58 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 ...
Harry Crimmins's user avatar
2 votes
0 answers
69 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,\...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
711 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 $...
user avatar
5 votes
2 answers
176 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 ...
Iosif Pinelis's user avatar
2 votes
1 answer
145 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}(...
ARG's user avatar
  • 4,342
1 vote
1 answer
158 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 ...
O. Richard's user avatar
1 vote
0 answers
101 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{...
Ozzy's user avatar
  • 383
3 votes
1 answer
3k 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. ...
Samrat Mukhopadhyay's user avatar
2 votes
1 answer
209 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 ...
avk255's user avatar
  • 543
1 vote
1 answer
92 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}}...
Meysam's user avatar
  • 21
1 vote
1 answer
154 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(...
litmus's user avatar
  • 91
1 vote
2 answers
218 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{...
liyan's user avatar
  • 33
4 votes
1 answer
146 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 ...
Collin's user avatar
  • 43
1 vote
0 answers
78 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 ...
Johan Wästlund's user avatar
7 votes
1 answer
434 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 ...
Erel Segal-Halevi's user avatar
4 votes
0 answers
141 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 ...
Rémi Peyre's user avatar
0 votes
0 answers
128 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 ...
Bernardo Recamán Santos's user avatar
9 votes
1 answer
460 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 $$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$ "...
Matthew Kahle's user avatar
1 vote
1 answer
104 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 ...
Iosif Pinelis's user avatar
63 votes
2 answers
3k 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 ...
Guillaume Aubrun's user avatar

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