# 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

**5**

votes

**2**answers

420 views

### An expansion from Ramanujan related to birthday problem

A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ...

**0**

votes

**0**answers

51 views

### Lipschitz function of independent subgaussian random variables

This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory).
If $X\in\mathbb{...

**0**

votes

**0**answers

62 views

### A closed form of mean-field equations

Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities)
$$P(q(t+\Delta t)-q(t)=1)=\...

**2**

votes

**0**answers

101 views

### Normal multivariate orthant probabilities

(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.)
Let $\mathbf{...

**6**

votes

**0**answers

111 views

### What can be said about percolation clusters after deleting a positive fraction of edges in general?

Start with a bond-percolation process just above criticality, say $p=1/2+\varepsilon$ on the graph $\mathbb Z^2$ with $\varepsilon>0$.
Sample $D\in\{0,1\}^E$ from an independent product measure ...

**0**

votes

**0**answers

71 views

### Computing the support of the equilibrium measure in Johansson's 1999 paper “Shape fluctuations and random matrices” in detail?

I am trying to compute the equilibrium measure for the Meixner ensemble on page 19 (on the arxiv version). The "details" of the computation are in Section 6, where he finds the equilibrium measure is ...

**1**

vote

**1**answer

68 views

### Sum of Square of the Eigenvalues of Wishart Matrix

Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...

**1**

vote

**0**answers

127 views

### Minimising an Integrated Relative Entropy Functional

Suppose I am given
A probability distribution on $\mathbf R^d$, with density $\pi (x)$.
A family of transition kernels $\{ q^0 (x \to \cdot) \}_{x \in \mathbf R^d}$ on $\mathbf R^d$, with densities $...

**5**

votes

**1**answer

212 views

### Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...

**0**

votes

**0**answers

45 views

### What is the probability that $X_i$ is the $k^{th}$ order statistic in consecutive trials?

Consider $n$ r.vs ${X_1, X_2,...,X_n}$. Each is i.i.d drawn from some distribution $f(.)$. What is the probability that $X_i$ is the $k^{th}$ order statistic in any two consecutive trials?

**13**

votes

**1**answer

540 views

### Large-n limit of the distribution of the normalized sum of Cauchy random variables

What is the large-n limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...

**3**

votes

**0**answers

83 views

### A lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution

Given a set of Bernoulli random variables $x_1, \dots, x_n$ (not necessarily identical) with $X= \sum_{0<i\leq n} x_i$, I am intrested in finding a lower-bound for $\frac{\mathbb{E} [ \min (X,k) ]...

**1**

vote

**0**answers

90 views

### Hashed coupon collector

The story:
A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection.
Every day, a random customer arrives and buys his favorite card (that is, each customer ...

**2**

votes

**0**answers

73 views

### Arithmetic structure of non-zero cumulants

It is known that any non-Gaussian distribution must have infinitely many non-zero cumulants (Marcinkiewicz). I was wondering if something stronger is known about the structure of non-zero cumulants. ...

**2**

votes

**0**answers

65 views

### Distribution of a linear pure-birth process' integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:
$$
Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg]
$$
...

**0**

votes

**1**answer

115 views

### Symbol for monotone relationship between two probability distributions

Motivation:
At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if ...

**3**

votes

**0**answers

75 views

### Probability of a random collection of subsets being a cover

Consider the set $[n]=\{1,2,\ldots,n\}$. Suppose for each set $A\subseteq [n]$ I have a $p_A \in [0,1]$. I now create a random collection $\mathcal{W}\subseteq\mathcal{P}([n])$ of subsets of $[n]$ by ...

**2**

votes

**0**answers

74 views

### Average number of pieces of a random piecewise-linear function

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...

**3**

votes

**1**answer

141 views

### Understanding Finite Size Scaling in Percolation Theory

Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...

**0**

votes

**0**answers

89 views

### Linear Independence of random binary vectors

Suppose we have $Y_1, \ldots, Y_n \in \mathbb{R}^m$, $n$ independent random vectors ($m \geq n$), where the entries of each $Y_i$ are i.i.d. Bernoulli random variables taking the values $\{0, 1\}$ ...

**6**

votes

**2**answers

218 views

### Slowest initial state for convergence of finite birth-and-death Markov chains

Consider the continuous-time birth-and-death Markov chain on $\{1,\cdots,n\}$ with all rates equal to $1$. Is it true that the convergence to equilibrium, in total variation distance, is slowest when ...

**0**

votes

**3**answers

144 views

### Reference request: book on stochastic calculus (not finance)

I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...

**9**

votes

**0**answers

249 views

### Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...

**5**

votes

**1**answer

133 views

### Ratio of integrals with increasing dimension over Euclidean balls

Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...

**3**

votes

**1**answer

123 views

### Lower bound of the expectation of the product of inner products of random vectors

I encountered the following value in my research:
Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$.
Denote
$$
L = \mathop{\mathrm{E}}_x[ \prod_{1\...

**6**

votes

**1**answer

91 views

### The distribution of the area of a region cut out by chordal SLE?

Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$.
For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...

**2**

votes

**0**answers

54 views

### Optimal tail bounds for the sample mean

Given a bounded random variable $X$ such that $|X| < M$, I want to know how many iid samples $x_1, \dots, x_n \sim X$ have to be drawn such that the sample mean $\bar{x} = (x_1 + \dots + x_n)/n$ ...

**5**

votes

**1**answer

96 views

### The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):
Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....

**2**

votes

**0**answers

55 views

### Bridging between Rosethal Inequalities and log convex tails

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$.
Then we have the classical "Rosenthal-type ...

**6**

votes

**1**answer

259 views

### Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...

**2**

votes

**0**answers

60 views

### Defining weak solutions to infinitely many SDEs on the same probability space

Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...

**0**

votes

**0**answers

10 views

### On statistics of perfect matchings between planar $4$ colorable and planar $3$ colorable

Does the mean for number of perfect matchings of random graphs that are planar and $3$ colorable much higher than graphs that are planar are not $3$ colorable?
Does a planar graph if $3$ colorable ...

**4**

votes

**1**answer

80 views

### Reference request: When is the variance in the central limit theorem for Markov chains positive?

I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/...

**3**

votes

**1**answer

110 views

### Maximal correlation and independence

Let $X$ and $Y$ be random variables. Then the maximal correlation $\rho_m(X;Y)$ is defined as
$$ \rho_m (X;Y) := \max_{(f(X),g(Y))\in S} \mathbb{E} [f(X)g(Y)] $$
where $S$ is the collection of pairs ...

**4**

votes

**0**answers

100 views

### For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?

Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...

**5**

votes

**1**answer

176 views

### Trace of inverse of random positive-definite matrix in high dimension?

Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...

**2**

votes

**1**answer

105 views

### Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance, with $p \in [1,\infty]$

Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...

**5**

votes

**1**answer

139 views

### Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...

**0**

votes

**0**answers

109 views

### An application of Girsanov's Theorem

Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, $a$ has adapted derivative, ...

**7**

votes

**1**answer

180 views

### Do i.i.d Sums Concentrate Any Faster Than Martingales?

Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae).
The simplest concentration inequality I ...

**6**

votes

**2**answers

315 views

### A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities.
ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$
I can only seem to find books/papers that list ...

**6**

votes

**1**answer

160 views

### Maxima of Brownian motion

It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s..
From a physics perspective it seems reasonable that when the disorder of the path of a ...

**2**

votes

**0**answers

78 views

### Joint drunkard walks

The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke.
My ...

**2**

votes

**0**answers

64 views

### How to obtain mathematical expectation with the vector as random variable?

In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...

**0**

votes

**1**answer

95 views

### Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...

**0**

votes

**0**answers

37 views

### Lp Wasserstein distance for two distributions which have equivalent partial marginals

Consider two distributions $p(x_1,x_2)$ and $q(x_1,x_2)$ which have equivalent partial marginals, say $p(x_1)=q(x_1)$. I am wondering if there is any relationship between two Wasserstein distances $W_{...

**5**

votes

**1**answer

180 views

### Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

The following result is well-known:
Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
$$H \left( (1 - \lambda)(x_1,y_1) + \...

**5**

votes

**0**answers

91 views

### Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...

**6**

votes

**1**answer

222 views

### Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...

**2**

votes

**1**answer

142 views

### Moment generating function of random unit vector

Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of
$$\mathbb{E}[\exp(X^Tv)]$$
for any $v$?