All Questions
9,497 questions
6
votes
1
answer
133
views
Coupling/Ordering of Brownian bridges
Suppose I have two 1D Brownian bridges $(B^{(1)}_t,t\in [0,1]),(B^{(2)}_t,t\in [0,1])$, one from $0$ to $0$ and one from $x$ to $y$ where $x,y \geq 0$. Is there a neat way to show that there exists a ...
- 228
2
votes
0
answers
104
views
Existence of Dirac measures in the context of joint and marginal distributions
Let $\pi$ be the joint law of $(X, Y)$ with marginal distributions $\mu$ and $\nu$. We assume that we have: for all $A \in \mathcal{B}(\mathbb{R})$ such that $\mu(A) > 0$
$$
\nu\left(\{y \in \...
0
votes
1
answer
85
views
Analytical approaches to approximate probability density functions of multivariate random functions
Given a random multivariate function $f(x, y, z)$, where $x, y, z$ are independent and identically distributed random variables with a probability distribution $\rho(X)$, I aim to approximate the ...
- 375
-1
votes
1
answer
80
views
Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
- 48.8k
3
votes
1
answer
74
views
Strong law of large numbers indexed by a directed set
Let $\xi_{1},\xi_{2},\ldots$ be a sequence of independent random variables with mean 0. For simplicity, assume that each $\xi_{i}$ only takes two values in $[-1,1]$.
Let $\mathscr{F}$ denote the ...
1
vote
1
answer
60
views
Reverse Doob’s maximal inequality for bounded martingales
Consider the set of discrete or continuous time $L^\infty$-bounded martingales $X$ with $X_0 = 0$ almost surely. Here $L^\infty$-bounded means $\|X\|_{\infty} := \sup_t \mathbb \|X_t\|_{L^\infty(\...
- 6,155
3
votes
1
answer
181
views
A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 6,155
1
vote
0
answers
48
views
Comparing two probability of connection in Bernoulli percolation on $\mathbb{Z}^2$
I want to know for bond Bernoulli percolation on $\mathbb{Z}^2$, does it holds that
$$ \mathbb{P} \left( (0,0)\longleftrightarrow (0,n) \right) \geq \mathbb{P} \left( (0,0)\longleftrightarrow (k,n) \...
- 95
5
votes
2
answers
557
views
A race to the bottom
Nate has a biased coin that comes up heads $\frac{1}{2} + \delta$ proportion of the time, where $0 < \delta \leq \frac{1}{2}$. He is competing against a large number $N$ people who each have fair ...
- 6,155
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
0
votes
1
answer
64
views
Sharpening Doob’s upcrossing inequality for Brownian motion
Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.
Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:
If $U(a,b)$ denotes the number ...
- 6,155
1
vote
0
answers
68
views
Bialgebras in 1/Kl(D)
$1/Kl(D)$ is the comma category of the one element set in the Kleisli category of the distribution monad. There is mention of it here. The objects are probability distributions called states and the ...
- 331
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
3
votes
1
answer
98
views
Error bound for MonteCarlo estimate of elements in Gram-Matrix
Suppose I have a $n\times n$-symmetric positive-definite matrix $A$ with elements:
\begin{align}
[A]_{ij}=\int_{\Omega}f_i(x)f_j(x) \, dx, \quad i,j=1,\ldots,n
\end{align}
where $\Omega\subset \mathbb{...
- 93
0
votes
2
answers
60
views
Do continuous martingales satisfy this nice terminal inequality?
Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
- 6,155
0
votes
0
answers
44
views
Large Deviation Principle for an adaptive sampling rule for Multi Armed Bandits
Consider the following adaptive strategy for sampling from a Multi Armed Bandit with $K$ arms:
Split the $T$ rounds into $N (\in \mathbb{N})$ disjoint intervals. Each interval is indexed by $i=1,2,\...
- 9
1
vote
0
answers
48
views
Quantile maximization of the difference of random constrained quadratic optimization problems
I am interested in understanding the family of parametrized random variables defined by the pushforward map
$$
\lambda_x : \varepsilon \mapsto \underset{z_1 \in \mathbb{R}^n :\, h^T z_1 = 0, \; z_1 \...
- 11
0
votes
0
answers
66
views
Long-time conditioning for a Markov Chain
I am studying MERW and for some reasons, i would like to know if, if I have $(X_n)$ an irreducible Markov Chain, I can say that
$\mathbb{P}(X_1=x | X_0=a, X_n = b)$ goes to $\mathbb{P}(X_1=x | X_0=a)$ ...
- 11
1
vote
1
answer
196
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
- 11
0
votes
0
answers
32
views
A question on Poisson approximation of number of secure rooks on a d-dimensional chessboard
This question was given in our first year undergraduate Probability I course.
In $d$ dimensions the lattice points $i = (i_1, i_2, \cdots, i_d)$ where $1\leq i_j\leq n$ may be identified with the “...
- 149
1
vote
1
answer
50
views
Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
- 23
5
votes
1
answer
183
views
What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 6,155
4
votes
1
answer
150
views
Convex order between Gamma distributions and Exponential distributions
Let $ (b_1, \dots, b_n) $ be a tuple of positive integers. Define independent random variables $ Y_i \sim \text{Gamma}(b_i, b_i) $ (shape and rate parameter both equal to $ b_i $) for $( i = 1, \dots, ...
- 55
2
votes
0
answers
71
views
Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
- 261
3
votes
1
answer
435
views
What is the connection between these three methods of generating this sequence?
I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
- 39
0
votes
0
answers
45
views
Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
- 111
0
votes
1
answer
99
views
Expressing a multivariate normal distribution as a mixture of uniform distributions?
Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
1
vote
1
answer
75
views
Probability of correctly guessing the maximum event probability of a multinomial distribution
I have a sample from multinomial distribution with $n$ trials, and $k=3$ options. I know that one of the event probabilities $p_i$ is larger than the two others (who are equal). I'm trying to guess ...
- 267
2
votes
0
answers
76
views
Inequalities concerning cummulative distributions of binomials
For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$...
- 121
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
0
votes
2
answers
134
views
Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 150
1
vote
0
answers
62
views
Convergence of iterated average Bayesian posterior to high entropy distribution
Setup
Assume $p_Y \in \Delta^n$ is a probability vector obtained by $p_Y=L_{Y|X}p_X$, where $L_{Y|X} \in \mathbb{R}^{n \times m}$ is an arbitrary likelihood (i.e, a column stochastic matrix) and $p_X \...
- 11
1
vote
0
answers
61
views
Bound on $\int_0^1\sqrt{\log N_{[]}(\varepsilon,\mathcal{F},d)} \, d\varepsilon$ over the class of half-spaces $\mathcal{F}$ on $\mathbb{R}^d$?
For a class of functions $\mathcal{F}$ and a pair $f,g\in\mathcal{F}$ with $f\leq g$, the interval $[f,g]=\{h:f(x)\leq h(x)\leq g(x),\forall x\in\mathbb{R}^d\}$ is called a bracket for $\mathcal{F}$. ...
- 141
3
votes
1
answer
116
views
Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
- 7,545
16
votes
0
answers
309
views
Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
- 3,507
2
votes
0
answers
92
views
Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
5
votes
0
answers
411
views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 573
2
votes
3
answers
338
views
Sum of RVs satisfying Bernstein condition on moments
Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that
$$
|\mathbb{E}[(X - \mu)^k]| \le \frac ...
- 367
5
votes
0
answers
68
views
Distribution of this integral of Fourier multiplier
In Barashkov and Gubinelli (2019) section 2, the authors make the claim that the distribution of $$Y_t = \int_0^t \langle D \rangle^{-1}\sigma_s(D)dX_s$$ is given by the pushforward $(\rho_t(D))_*\...
- 51
3
votes
1
answer
218
views
Pathwise linearization of diffusion processes
Let $W$ be a standard $n$-dimensional Brownian motion, and $X$ the diffusion process given by the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $\mu: \mathbb R^n \to \...
- 6,155
5
votes
0
answers
112
views
Discrete random walk in an expanding cage (i.e. in a growing domain)
In the book "A guide to First-Passage Processes" by Sidney Redner, a section is dedicated to the survival probability of a random walker in a growing domain.
For a fixed-length interval $[0,...
- 274
1
vote
1
answer
215
views
Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
- 21
2
votes
0
answers
93
views
$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE
$$\partial_t u=\Delta u-u^3+\xi,$$
where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
- 1,904
2
votes
0
answers
58
views
Inclusion-Exclusion formulae for number of SAWs on $\mathbb{Z}^d$ of length $n$ [closed]
Here is my attempt at lower bounding the number of SAWs on $\mathbb{Z}^d$ of length $n$: In $\mathbb{Z}^d$, consider the $2^{d-1}$ lines of the form $\epsilon_1 x_1 = \epsilon_2 x_2 = \epsilon_3 x_3 \...
- 21
0
votes
0
answers
102
views
Formalizing the "pseudorandomness" of primes
Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
- 1
3
votes
1
answer
135
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75
2
votes
0
answers
83
views
Random time change and ergodicity
I guess it is a standard question in ergodic theory but I failed to find any reference to similar problems and I have no clue on how to tackle it.
Let $(B_{t})_{t\in \mathbb{R}}$ be a standard ...
- 21
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
1
vote
0
answers
66
views
Confusion about central limit theorem by Chandrasekhar et al
In this paper by Chandrasekhar et al. (General Covariance-Based Conditions for Central Limit Theorems with Dependent Triangular Arrays) we can find the following CLT (for simplicity paraphrased in ...
- 677
4
votes
1
answer
110
views
Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
- 153