**0**

votes

**0**answers

5 views

### Probability of an event based on percentage in fixed lapse of time

I am a software engineer. I am also a former triathlete that rides with a large group of friends every time we have a chance.
i am trying to come up with a little software to distribute among us ...

**1**

vote

**1**answer

59 views

### If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space.
Usually, we have a ...

**7**

votes

**1**answer

311 views

### Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...

**4**

votes

**2**answers

103 views

### Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...

**4**

votes

**1**answer

79 views

### Negative population variable importance

I asked this question on stats.stackexchange and even elsewhere, but it never received an answer.
I just state the probabilistic problem here. It is about the optimality of the conditional ...

**0**

votes

**1**answer

103 views

### Proving an inequation based on binomial distributions

Problem statement
Let $c \in \mathbb{N}$, $n_1 \in \mathbb{N}_0$, and $n_2 \in \mathbb{N}_0$ be integers and $p$ a probability. Furthermore, let $b(m,j,p) = \binom{m}{j}p^j(1-p)^{m-j}$ denote the ...

**2**

votes

**0**answers

47 views

### Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
...

**1**

vote

**0**answers

36 views

### How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$
where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$
Initial condition:
\begin{eqnarray}
F(0, r) & = ...

**0**

votes

**0**answers

18 views

### Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map
$\theta \to I(\theta)$ where $I(\theta)$ is the ...

**3**

votes

**1**answer

63 views

### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...

**2**

votes

**1**answer

65 views

### Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...

**2**

votes

**1**answer

130 views

### How many times does a simple symmetric random walk of length n return to the origin?

Consider the simple symmetric random walk on the integers starting from
the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as
$$ w:= \omega_0 \omega_1 \ldots ...

**2**

votes

**1**answer

92 views

### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...

**-4**

votes

**0**answers

42 views

### Probability Inequality when X > Y > 0 [on hold]

I want to know whether the following statement is true or not, and the proof.
Let X, Y be random variable, satisfying X > Y > 0, and have finite variance, $Var(X) < \infty$ and $Var(Y) < ...

**4**

votes

**3**answers

171 views

### Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
...

**4**

votes

**1**answer

213 views

### 'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...

**1**

vote

**0**answers

52 views

### Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...

**-4**

votes

**1**answer

51 views

### Win/Lose ratios and selection [on hold]

Imagine a following scenario:
You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...

**-6**

votes

**0**answers

25 views

### Trouble with an assignment [on hold]

Can anyone please be kind enough to help me with this.
On one shelf there are 5 hardcover books and 6 paperbacks and on the other shelf there are 7 hardcover and 4 paperback. From the first shelf ...

**0**

votes

**0**answers

52 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

**2**

votes

**0**answers

59 views

### Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.
So $G\in\mathcal{G}$ can be written ...

**0**

votes

**1**answer

130 views

### Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) ...

**7**

votes

**1**answer

473 views

### Doob Martingale: Where is the catch?

I am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support.
I am attempting to use the method of bounded ...

**0**

votes

**0**answers

47 views

### continuous vs discrete random walk [on hold]

For 1D random walk in discrete case the probability $P_N(X)$ of finding walker at position $X$ after $N$ steps has a binomial distribution, moreover when $N+X$ is odd then probability is 0.
Let's ...

**-3**

votes

**0**answers

26 views

### Statistics, probability [on hold]

Eight of the 40 newly built cars are selected at random to be checked for steering defects. Suppose 10 0f the cars have such defects. What is the probability that all 8 of the selected cars have ...

**2**

votes

**1**answer

234 views

### Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...

**0**

votes

**3**answers

163 views

### Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which
$$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent
I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...

**1**

vote

**1**answer

77 views

### Spacing of the largest singular values of Wishart matrix

Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...

**11**

votes

**2**answers

208 views

### What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdős-Rényi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Albert ...

**1**

vote

**1**answer

942 views

### A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version:
Randomly assign (with replacement) $N$ balls to $M$ urns. ...

**1**

vote

**1**answer

221 views

### Correlation between two distance measures on bitstrings

I have an infinite collection of $0/1$ random strings of length $n$ (i.e., say 010001110101), where each digit is an independent Bernoulli RV, with parameter $p_i$, $i:1...n$.
Define the "trait ...

**7**

votes

**1**answer

213 views

### Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...

**0**

votes

**0**answers

35 views

### Maximal Correlation with Weak Gaussian Perturbation

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...

**2**

votes

**1**answer

168 views

### What is the formal name of this set-related concept?

I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name?
Let $S$ be a family of finite sets.
...

**1**

vote

**0**answers

35 views

### Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix ...

**2**

votes

**1**answer

254 views

### Inequality for square of the subgaussian distributions

Hi all,
For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically:
Let $a$ be unit vector in ...

**0**

votes

**0**answers

22 views

### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...

**22**

votes

**1**answer

694 views

### Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...

**4**

votes

**1**answer

181 views

### Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...

**7**

votes

**1**answer

865 views

### Van Den Berg-Kesten-Reimer inequality

Van Den Berg-Kesten-Reimer inequality
For a given positive integer $n$ and for every $i\in[n]$, denote by $\mu_i$ a probability measure on a finite set $\Omega_i$. Call $\mu$ and $\Omega$ the ...

**18**

votes

**5**answers

4k views

### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

**1**

vote

**0**answers

106 views

### Linking Wasserstein and total variation distances

I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...

**0**

votes

**0**answers

43 views

### How to check numerically iterated logarithm law ? (How to choose cutOff lim_n sup_{m: n<= m<= CutOff} ) ?

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then
$$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...

**0**

votes

**0**answers

59 views

### Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...

**1**

vote

**1**answer

54 views

### Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...

**6**

votes

**2**answers

231 views

### 2-Wasserstein (optimal transport) and extension to the set of all signed measures

Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...

**2**

votes

**0**answers

100 views

### Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$.
It's well known that this process is determinantal if one can find a positive semidefinite matrix $K$, ...

**0**

votes

**1**answer

80 views

### Supremum of a martingale

Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = ...

**0**

votes

**1**answer

108 views

### Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...

**-1**

votes

**0**answers

8 views

### Hypothesis testing for independent and non-identical distribution [migrated]

I want to apply the hypothesis for my problem. According to A. Wald regarding sequential hypothesis, he used independent and identically distributed (iid) observations or samples but in my case my ...