# 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.

7,279
questions

**7**

votes

**1**answer

265 views

### Expectation for game choosing uniformly number in $[0,1]$ until it decreases

We are playing a game where we keep on choosing a number from the uniform distribution U(0,1). The game goes on until we have the current number less than the previously picked number, i.e. the game ...

**5**

votes

**1**answer

107 views

### Minimization of an entropy type functional

Let $\mathcal P$ be the set of probability densities on $[0,1]$ with mean $1/2$, i.e. $p\in \mathcal P$ iff
$$\int_0^1 p(x)dx=1,\quad \int_0^1 xp(x)dx=\frac{1}{2}\quad \mbox{and}\quad p(x)\ge 0, ~~\...

**11**

votes

**1**answer

197 views

### (Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$.
In the case of bounding $E(XY)$...

**0**

votes

**1**answer

41 views

### How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme?

I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is ...

**2**

votes

**0**answers

68 views

### Conditional probability of maximum and minimum of Brownian motion

I want to ask for the following problem. Let $(W_t)_{t\geq 0}$ be the standard Brownian motion. For each $t>0$, we call $$m_t =\inf_{0 \leq s \leq t} W_s, \qquad M_t = \sup_{0 \leq s \leq t} W_s.$...

**1**

vote

**1**answer

199 views

### Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...

**1**

vote

**0**answers

61 views

### Number of $k$-cycles in a random Mallows permutation

It is well-known that for uniform random permutations, the number of $k$ cycles for fixed $k$ is distributed as a Poisson random variable with mean $1/k$.
I am looking for similar results on the ...

**2**

votes

**0**answers

79 views

### Convergence of Gibbs distribution to Dirac measure [closed]

Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$:
$$
q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right).
$$
Denote $x^* = \arg \...

**0**

votes

**0**answers

150 views

### Unexpected autocorrelations in sequence of primes modulo 4

It is well known that there is a little bias in the distribution of prime residues modulo 4. But the bias eventually vanishes. I looked at the first million primes, and the counts are as follows:
...

**-1**

votes

**0**answers

35 views

### Sampling from characteristic/moment generating function in higher dimensions

The question is - if I am given a characteristic function in $\mathbb{R}^d$ and I want to sample from this implicit distribution, how can this be done? I found work done by Devoye in the 80s and ...

**2**

votes

**1**answer

29 views

### Matrix-valued cumulant generating function for Wishart matrices

Suppose we have an axis-aligned Gaussian vector $v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$, and consider the Wishart matrix $W = vv^\top$.
Is there a simple closed form/"Lowener order ...

**1**

vote

**1**answer

93 views

### Generating iid random vectors such that the distribution of their dot product is $\mathit{Uniform}[a, b]$

Take two independent and identically distributed random vectors $X_i$, $X_j$.
I want to find a multivariate distribution for these vectors such that the dot product $X_i^\top X_j \sim U[a, b]$.
This ...

**1**

vote

**1**answer

163 views

### Generalized random harmonic series

Let $Z_n=\sum_{k=1}^n a_k X_k$ with $(a_k)$ a strictly decreasing sequence of positive real numbers that tend to zero. The random variables $X_k$ are independent and satisfy $P(X_k=1) =p_k, P(X_k=-1)=...

**2**

votes

**0**answers

129 views

### Are there sets in the unit cube that cannot be in the domain of any finitely-additive, isometry-invariant probability measure?

The Vitali construction implies (given choice) the existence of a set such that for any translation-invariant, countably additive probability measure on $[0,1]$, that set is nonmeasurable and has ...

**1**

vote

**0**answers

188 views

### Measures on the plane and on $\mathbb{R}^3$

Let $\Delta$ be the 2-dimensional probability simplex. Denote a generic vector in $\mathbb{R}^3$ as $(x,y,z)$.
Can you find some $(a,b)\in \mathbb{R}^2$ and $(p_1,p_2,p_3)\in \Delta$ such that ...

**0**

votes

**0**answers

36 views

### Local differentially private normal vectors

We're given a vector $x\in \mathbb{R}^d$ whose coordinates where sampled from a known normal distribution $\mathcal{N}(0, \sigma^2)$.
How should I send this vector while maintaining (local) ...

**3**

votes

**1**answer

97 views

### Top singular value of large random matrices: concentration results

Let $A$ be a $n\times m$ random matrix, whose elements $a_{ij}$ are independent standard Gaussian random variables.
I am interested in the case $n=\alpha N\,$, $\,m=(1-\alpha)N$ for $\alpha\in(0,1)$ ...

**2**

votes

**1**answer

104 views

### Reverse martingale convergence theorem in Banach spaces

In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...

**0**

votes

**1**answer

53 views

### The distribution of number of reverse order pairs in a randomly permuted array

There is an array $a_1,\dotsc,a_n$ whose elements are pairwise distinct. We define a reverse order pair to be an ordered pair $(a_i,a_j)$ such that $i < j$ and $a_i > a_j$. Consider the total ...

**0**

votes

**1**answer

50 views

### Iterated integrations by parts using the fractional Laplacian

Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...

**-1**

votes

**0**answers

65 views

### If $X$ is a Lévy process and $Y_t:=X_{s+t}-X_s$, do $(\Delta Y_t)_{t\ge0}$ and $(\Delta X_t)_{t\ge0}$ have the same distribution?

Let $(X_t)_{t\ge0}$ be a càdlàg Lévy process with $X_0=0$ and $$X_{t-}:=\lim_{s\to t-}X_s$$ and $$\Delta X_t:=X_t-X_{t-}$$ for $t\ge0$.
If $s\ge0$, we can show that $$Y_t:=X_{s+t}-X_s\;\;\;\text{for }...

**4**

votes

**1**answer

259 views

### Irreducible representations of U(n) and probability of being close to having fixed points

Suppose $\pi:U(n)\rightarrow GL(V)$ is a positive-dimensional irreducible representation of the unitary group. Given $\varepsilon>0$, how could one rigorously show that the probability that $|\det(...

**0**

votes

**0**answers

28 views

### Conditions on $g$ which ensure the function $\phi(t):=\int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,\mathrm{d}\theta$ is $C^k$ on $(-1,1)$

Given an almost-everywhere continuous function $g:[-1,1] \to \mathbb R$, define another function $\phi_g:[-1,1] \to \mathbb R$ by
$$
\phi_g(t) := \int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,...

**0**

votes

**1**answer

71 views

### Gradient of Wasserstein distance in the sense of Otto's calculus

I am learning the idea of "gradient" of a functional in Otto's calculus. It is defined as follows.
Suppose the space we are thinking about is $(\mathcal{P}_{2,AC}(\mathbb{R}^d),W_2)$, the ...

**0**

votes

**1**answer

63 views

### Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost

I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from ...

**3**

votes

**1**answer

119 views

### Limiting eigenvalue distribution of $(I-A)^T(I-A)$

Let $A\in\mathbb R^{n\times n}$ be a random Gaussian matrix with i.i.d entries from $\mathcal N (0, \frac{a}{\sqrt{n}})$. By Marchenko-Pastur we know the limiting distribution of the eigenvalue of $A^...

**1**

vote

**1**answer

176 views

### Construct a random vector as a function of another random vector

ASSUMPTION 1: there exists a continuous random vector $(X,Y,Z)$ such that
$$
\begin{cases}
p_1=\Pr(X\geq 0, Z\geq 0)\\
p_2=\Pr(Y\geq 0, Z< 0)\\
p_3=\Pr(X< 0, Y<0)\\
\end{cases}
$$
where $(p_1,...

**4**

votes

**1**answer

89 views

### Wasserstein-type concentration inequalities for empirical measures on polish spaces

Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...

**0**

votes

**2**answers

132 views

### Couplings as generalized functions

I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $\mu$ and $\nu$ on a metric space $X$ by
$$
W_p(\...

**2**

votes

**1**answer

54 views

### Inequality for increments of $r$th absolute moments of martingales, $1<r<2$

If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that
$$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...

**2**

votes

**0**answers

75 views

### Limiting PDF of the eigenvalue of random Gaussian matrix

It has been proven that the CDF of the eigenvalue distribution of random Gaussian matrix converges to a uniform disk circular law. Is it true for the PDF of the limiting eigenvalue distribution? In ...

**4**

votes

**1**answer

97 views

### Expected even power of absolute value of an element of a random unitary matrix

Let $t$ be a natural number. For a unitary matrix $U$ let $U_{1,1}$ be the top left matrix element of $U$. I am trying to figure out the value of $\int |U_{1,1}|^{2t} dH(U)$ where $H$ is the Haar ...

**2**

votes

**1**answer

61 views

### On the marginal distributions of an absorbed diffusion

This question can be seen as a variant of the post Bounded density for diffusions with diffusion coefficients bounded away from $0$ by Iosif Pinelis. Namely, consider the diffusion
$$X_t=\int_0^t a(s,...

**5**

votes

**1**answer

273 views

### Definition of infinite-dimensional Gaussian random variable

For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:
Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random
variable $u \in H$ is ...

**2**

votes

**0**answers

82 views

### Probability of a finite cylinder set in a free group

Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...

**0**

votes

**1**answer

48 views

### Almost sure convergence of the supremum over a class of random variables

Let $\mathcal{X}_n=\{ X_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\...

**4**

votes

**3**answers

142 views

### Probability that $k$ random subsets of a fixed size covers a set

Let $A=\{1,\ldots,n\}$. Now, we uniformly randomly select $k$ subsets, $A_i$ of size $d$ from $A$. What is the probability that $\bigcup_i A_i=A$? This seems to be natural variant of the set cover ...

**4**

votes

**2**answers

191 views

### Do these distributions have a name already?

In playing with some math finance stuff I ran into the following distribution and I was curious if someone had a name for it or has studied it or worked with it already.
To start, let $\Delta^n$ be ...

**0**

votes

**0**answers

26 views

### finite-rank PSD pertubation to spectral gap of top eigenvalues

Suppose that $A\in\mathbb{R}^{n\times n}$ is a deterministic PSD matrix and $G\in\mathbb{R}^{n\times k}$ is i.i.d. $N(0,1)$ matrix, where $k\ll n$ is a positive integer. Let $B = A + GG^T$. I want to ...

**1**

vote

**1**answer

139 views

### What is this optimization problem called

Let $X$ be a set and $\mathcal{F}$ be a set of functions $f:X \to \Bbb{R}$ (for my purposes, it is fine to assume both sets are finite).
For a probability distribution $\mu$ on $\mathcal{F}$, we ...

**1**

vote

**0**answers

65 views

### A formula involving the heat kernel on the universal cover of a punctured plane

I am looking for the earliest reference to the following formula:
$$
\int_0^\infty\tilde{P}(1,e^{i\alpha},t)\frac{dt}{t}=\frac{1}{\pi \alpha^2},\quad \alpha>0,
$$
where $\tilde{P}(x,y,t)$ is the ...

**4**

votes

**2**answers

176 views

### Wasserstein convergence of "series expansion'' of probability measure

Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...

**3**

votes

**1**answer

95 views

### Reference request: probabilistic models on climate (change)

I am looking for probabilistic models to address climate change. Are they known in the existing literature?
I have found the post Math behind climate modeling. concerning PDE models.
Many thanks for ...

**1**

vote

**0**answers

96 views

### A question on Gaussian small ball probability

Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$
where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...

**1**

vote

**2**answers

64 views

### Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$

Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (-1-t_0,1-t_0)$, ...

**0**

votes

**1**answer

92 views

### CLT for bounded difference functions

Let $X_1, \ldots, X_n$ be independent and identically distributed random variables. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a bounded difference function, i.e., for any $x,y \in \mathbb{R}^n$ that ...

**6**

votes

**2**answers

221 views

### Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...

**1**

vote

**1**answer

70 views

### Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question.
Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....

**0**

votes

**2**answers

72 views

### What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...

**2**

votes

**3**answers

202 views

### Dominated convergence theorem when the measure space also varies with $n$

Let $(f_n)_n:X \to \mathbb R$ be a sequence of measurable functions on a measurable space $X$ converging pointwise to a function $f:X \to \mathbb R$, and let $(\mu_n)_n$ be a sequence of finite ...