**0**

votes

**0**answers

10 views

### What's the probability that at least one of k top data still in top k positions of a data set with error?

Let {${d_1,d_2,d_3,..., d_k,...d_n}$} is a descendingly sorted data set.
Now we suppose that each data in the set has a probability $p_e$ to go wrong.
The problem is what's the probability that at ...

**0**

votes

**0**answers

18 views

### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...

**-7**

votes

**0**answers

47 views

### A true-false exam has five questions. Andy is completely ignorant and so he tosses a fair coin to answer each question [on hold]

A true-false exam has five questions.
Andy is completely ignorant and so he tosses a fair coin to decide his answer to each question.
What is the probability that he scores at least four correct?

**1**

vote

**0**answers

20 views

### Asymptotic variance for partial sum of a stationary process

Let $X = (X_1, \dots, X_n, \dots)$ be a sequence of random variables. We assume that the process X is stationary i.e. for any integer $k$, any set of indices $i_1 < \dots < i_k$ and any integer ...

**-4**

votes

**0**answers

55 views

### Random Walk on the Natural Numbers [on hold]

Let's consider a random walk on the natural numbers (1,2,3...), where we always go from 1 to 2 and otherwise to the left with probability p, hence to the right with probability 1-p.
I aim to find ...

**1**

vote

**0**answers

32 views

### Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem:
Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$.
Let $a( x;. )$ and $f(x;.)$ be ...

**3**

votes

**0**answers

83 views

### Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...

**2**

votes

**0**answers

39 views

### Restricted singular values of random matrix

Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law.
Now let's ...

**1**

vote

**0**answers

75 views

### First passage time of a pure drift process

I am facing the following unusual problem: $Z_t$ is a pure drift process of the form
$$ dZ_t = \kappa(X_t - Z_t) dt $$
where $X_t$ is another bounded process.
I am interested in computing / ...

**0**

votes

**0**answers

35 views

### What is the Vapnik-Chervonenkis dimension of sigmoidal functions? [migrated]

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...

**5**

votes

**1**answer

248 views

### Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...

**0**

votes

**1**answer

243 views

### About an integral equation

I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...

**4**

votes

**1**answer

168 views

### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

**5**

votes

**2**answers

276 views

### Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$.
Our goal is to find $t\in[0,1]$ such that
$$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$
Is there a closed-form ...

**0**

votes

**1**answer

52 views

### Probability spaces involved in using Bayesian Inference

I am currently reading "Statistical and Inductive Inference by Minimum Message Length" by C.S. Wallace. In this, Wallace gives a fairly informal account of Bayesian Inference which, in the case ...

**3**

votes

**0**answers

73 views

### Ask for reference of a stochastic process

I would like to know whether the following stochastic process is well studied.
Let $\{U_k: k \ge 1\}$ be a sequence of i.i.d random variable. $U_1$ is uniformly distributed on the unit interval $[0, ...

**-1**

votes

**0**answers

49 views

### Probability: short circuiting amongst recycled batteries [closed]

I am a student at University of Illinois with some batteries to recycle. The student union collects batteries, but requires the terminals to be taped. I think this makes a lot of sense for 9 volt ...

**-3**

votes

**3**answers

267 views

### Determinant of matrix from set {-1, 1}

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...

**6**

votes

**0**answers

136 views

### Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...

**1**

vote

**1**answer

123 views

### How does Azuma's Inequality result from Pinelis Inequality?

According to [1]
Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
$S(\mathcal{X})$ denote the class of all sequences
$f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...

**1**

vote

**0**answers

38 views

### Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...

**1**

vote

**2**answers

125 views

### Speed and absence of non-constant bounded harmonic functions

For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...

**0**

votes

**0**answers

83 views

### Best possible concentration inequality in high dimensions

Let $X_1,\ldots,X_n$ be independent random variables in $\mathbb{R}^d$ with $EX_i=0$ and $||X_i||_{2}\leq 1$. What is the best known exponential upper bound for
$$P(||X_1+\cdots+X_n||_{2}>x)?$$
In ...

**2**

votes

**0**answers

70 views

### McDiarmid's inequality on normed spaces

McDiarmid's inequality says if a function $f: \mathcal{X}^n\to\mathbb{R}$ has the property that
$$
\sup_{x_1,\dotsc,x_n,x'_i\in\mathcal{X}} ...

**3**

votes

**1**answer

109 views

### Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let ...

**5**

votes

**0**answers

144 views

### A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...

**5**

votes

**1**answer

349 views

### Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...

**0**

votes

**1**answer

56 views

### Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?

Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine ...

**1**

vote

**0**answers

52 views

### Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of
$$
dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0
$$
where $B$ is a standard 1-dimensional Brownian motion.
$X$ can be viewed as the norm of a ...

**8**

votes

**1**answer

213 views

### Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...

**8**

votes

**0**answers

115 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**2**

votes

**1**answer

159 views

### Random walks with exponential decreasing steps

Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking ...

**0**

votes

**0**answers

48 views

### Expected number of distinct nodes visited in a directed bipartite graph [migrated]

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$.
All the edges start from a vertex in $I$ and end on a ...

**4**

votes

**1**answer

218 views

### A central limit theorem for a trigonometric series involving primes

In some recent work I found I needed to prove a central limit theorem for
the interesting series:
$\sum_{n=1}^\infty \cos (u \log p_n) $
where u is a random variable uniform on the interval ...

**1**

vote

**0**answers

39 views

### Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...

**4**

votes

**1**answer

89 views

### Earth mover/Wasserstein distance between a pdf and an empirical distribution

This question is inspired by this much older question:
Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a continuous probability distribution on a compact subset of ...

**0**

votes

**2**answers

151 views

### “Convolution” for Multiplying Random Variables

The following situation arises frequently in probability.
Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...

**2**

votes

**0**answers

54 views

### Controlling fluctuations in a Markov chain

For $N>0$, consider the Markov chain $x_n$ on $\{0,1/N,...,1\}$ that moves up by $1/N$ at rate $(c-x_n/2)N$ and down by $1/N$ at rate $(c+x_n/2)N$. As $N\rightarrow\infty$ sample paths approach ...

**0**

votes

**1**answer

48 views

### Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that
$$\mathbb E[X_iX_j] = \delta_{ij}.$$
Now take $Z\in L^2(\Omega, \mathbb P)$ and define ...

**0**

votes

**0**answers

18 views

### Literature on the notion of combining two discrete stationary processes with the latter process slowed down

Is there any literature about the following way of combining two stationary processes?
Let $X_1, X_2, \dots$ be a discrete-time stationary process. Let $A$ be a subset in its sample space. Let ...

**1**

vote

**1**answer

83 views

### Hoeffding's inequality for vector valued random variables

Is there a version of Hoeffding's inequality for vector valued random variables?
This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...

**1**

vote

**1**answer

74 views

### Mixing time of a continuous time Markov chain with arbitrary rate matrix

I would like to calculate the mixing time of a continuous time starting from the rate matrix and not necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite ...

**1**

vote

**1**answer

134 views

### Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let
$p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$.
I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...

**5**

votes

**2**answers

211 views

### Regularity of random Fourier series

The following two statements appear to be true (but do correct me if I am wrong):
The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm ...

**3**

votes

**0**answers

56 views

### Probability that a random projection doesn't reduce the distance of a point from a subspace too much

Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; ...

**7**

votes

**5**answers

433 views

### Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...

**1**

vote

**1**answer

202 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**3**

votes

**1**answer

103 views

### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$):
$$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$
where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...

**2**

votes

**0**answers

51 views

### Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...

**6**

votes

**0**answers

127 views

### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...