# 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.

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

7,901
questions

3
votes

1
answer

156
views

The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?

0
votes

2
answers

114
views

Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...

1
vote

0
answers

50
views

We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...

8
votes

5
answers

483
views

Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$.
So $D$ has some probability ...

1
vote

0
answers

122
views

Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider
$$\sum_i x_i M_i$$
for random variables $x_i$.
What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$
...

2
votes

1
answer

214
views

Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation,
$$e(A|x) = \int_X e(...

0
votes

0
answers

27
views

This is motivated by a specific question here
Suppose $X(s)$ is a Gaussian field in $[0,1]$ with some short range correlation. Consider the integral
$$\int E\left[\prod_{i\geq 1} (e^{X(a_{i}-c)}-1)\...

3
votes

0
answers

123
views

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...

3
votes

0
answers

92
views

Let $(a_j)_{j \in \mathbb{N}_0}$ be a real-valued sequence such that $\sum_{j = 0}^\infty a_j^2 < \infty$.
Further, define an infinite moving average time series $X = \{ X(t), t \in \mathbb{Z}\}$ ...

5
votes

2
answers

467
views

There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...

0
votes

0
answers

62
views

Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...

3
votes

1
answer

244
views

Suppose we know two sets of distributions
$A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$.
We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$.
...

1
vote

1
answer

106
views

Levy's Lemma asserts Lipschitz functions of vectors chosen uniformly from the unit hypersphere concentrate:
Lemma.
Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipschitz on the unit hypersphere. ...

0
votes

0
answers

72
views

Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables.
Let's say one can show that $E[X]^k \...

8
votes

2
answers

525
views

Why is there so much interest in the Markov categories which are Kleisli categories for monads corresponding to distributions etc. but not much discussion of the E.M. categories?
For example, the E.M. ...

4
votes

1
answer

228
views

Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...

0
votes

0
answers

83
views

We have that for a real valued random variable $X$,
$$
P(X=0) \leq \frac{\text{Var}(X)}{\left(\mathbb{E}(X)\right)^2}
$$
known as Chebyshev's inequality. Consider a random variable $X \in \{0,1,2,\...

2
votes

1
answer

220
views

Consider the following, overly simplified, model for determining an optimal wealth distribution for society:
Let $X$ be a random variable, which will model the distribution of wealth in a society.
The ...

1
vote

1
answer

88
views

Let $n,d,s$ be positive integers with $s \le d$, and let $B_0(d,s)$ be the set of all (real) $d$-dimensional vectors with at most $s$ nonzero components. Given an $n \times d$ matrix $X$ with rows $...

4
votes

1
answer

365
views

In Section 4.2.4 of [1], the authors write
In this section we consider a causal linear process
$$
X_t = \sum_{j = 0}^\infty a_j \varepsilon_{t - j}, \quad t \in \mathbb{N},
$$
where, without loss of ...

2
votes

1
answer

135
views

Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...

1
vote

0
answers

52
views

I am trying to come up with a generalisation of the Kelly formula for optimal fractional betting but and have hit a roadblock. The Kelly criterion is usually explained via a game that ends in 1 of 2 ...

0
votes

1
answer

123
views

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.
What are some nontrivial bounds on
$p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?
This is motivated by ...

2
votes

0
answers

73
views

For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...

7
votes

1
answer

233
views

Let $\epsilon <1/2$. Let $X$ be a random variable in $\mathbb Z$ such that $\mathbb P (X=x)\le \epsilon $ for any $x\in \mathbb Z$ (you may add any moment or regularity conditions on $X$ if needed)....

7
votes

1
answer

226
views

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...

5
votes

2
answers

165
views

Let $X$ be a random variable with probability mass function $p(x) = \mathbb{P}[X = x]$.
I know entropy $H(X)$ of $X$ measures the uncertainty of $X$ and
a large value of $H(X)$ means $p(x)$ is nearly ...

1
vote

1
answer

157
views

Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of:
$$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...

3
votes

1
answer

136
views

Suppose that $D_{KL}(p_1\parallel q)<1$ and $D_{KL}(p_2\parallel q)<1$. I'm trying to show that either $D_{KL}(p_1\parallel p_2)$ or $D_{KL}(p_2\parallel p_1)$ will have an upper bound close to ...

8
votes

0
answers

207
views

Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...

2
votes

1
answer

79
views

Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...

6
votes

0
answers

94
views

What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...

1
vote

0
answers

82
views

In an Erdős–Rényi random graph $G(n,p)$, the giant component emerges with thresholding function $p(n) = c/n$, where $c>1$.
When $c=1$, and $\lambda \in \mathbb{R}$, we can write or "...

0
votes

1
answer

99
views

After eating a bag of liquorice allsorts in one sitting, as one does, I noticed that it had contained an unusual amount of brown ones (which, you will agree, are an abomination that should never have ...

0
votes

0
answers

36
views

Because Markov categories are monoidal, their morphisms can be seen as boxes with one or more input wires and one or more output wires. If the Category supports deleting wires by marginalization or ...

3
votes

2
answers

164
views

I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...

1
vote

2
answers

96
views

Lets say $Y=\frac{1}{n}XX^\intercal$ and $X$ is a $n\times m$ random matrix whose entries are i.i.d gaussian. We know when $n$ and $m$ go to infinity with a fixed ratio, the singular values of $Y$ ...

0
votes

1
answer

99
views

Let $\{X_i, i \in \mathbb{N}\}$ be a sequence of non-lattice i.i.d. centered random variables, $\mathbb{E} |X_1| ^3 < 0$. Let $S_n = \sum\limits _{i=1} ^n X_i$ be the corresponding random walk and ...

1
vote

1
answer

164
views

In "Sharp threshold phenomena in statistical physics", H. Duminil-Copin, Japanese J. of Math. 14, 2019, a sharp transition of a boolean function is defined as follows:
A sequence of ...

0
votes

0
answers

51
views

We are given $n$ bins and $dm$ colored balls. There are $m$ colors in total such that there are exactly $d$ balls of each color. For each color, we randomly choose $d$ bins and put the $d$ balls of ...

0
votes

1
answer

153
views

Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...

1
vote

1
answer

80
views

I am currently reading "Smoothing of Multivariate Data" by Klemela. It contains Lemma 11.6, which upper and lower bounds the KL-divergence of two densities in terms of the $L_{2}$-metric. ...

6
votes

1
answer

304
views

Consider simple random walks that stop when reaching a given node $x$ in an undirected, unweighted and connected graph on $n$ nodes.
Let
$H(i,x)$ denote the (expected) hitting time from $i$ to $x$, ...

8
votes

1
answer

297
views

Since this question is on the front page again, a generalization.
Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides ...

0
votes

1
answer

156
views

Let's take a simple random walk on $\mathbb{Z}$, $(S_n)_{n\geq0}$, started at zero. If $\tau^+_0 = \inf\{n \geq 1: S_n = 0\}$ is the first time the walk returns on zero, we know that $\mathbb{E}[\tau^+...

0
votes

1
answer

141
views

I am reading a book on stochastic processes. The author proved Itô formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Itô formula for $x(t)=a(t)+b(t)...

3
votes

1
answer

67
views

Traditionally, stochastic dominance is defined using the cumulative distribution function(CDF). But sometimes, the CDF is not easily to be obtained. For example, the generalized noncentral Chi-square ...

2
votes

0
answers

107
views

I am looking for an upper bound on the following integral:
$$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$
...

1
vote

0
answers

104
views

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...

3
votes

1
answer

414
views

It is known that
$$
\cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1})
$$
Is it true that
$$
f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\...