# 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.

8,494
questions

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votes

1
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63
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### Why do we define independence for zero-probability events?

I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of ...

0
votes

0
answers

160
views

### Is the integer factorization into prime numbers normally distributed?

Let $P_1(n) := 1$ if $n=1$ and $\max_{q|n, \text{ }q\text{ prime}} q$ otherwise, denote the largest prime divisor of $n$.
Let us define some rooted trees $T_{n,m}$ for $1 \le m \le n$ by:
$T_{n,m}$ ...

2
votes

1
answer

87
views

### Does $L^1$ boundedness and convergence in probability imply convergence in probability of the Cesaro sums?

Let $X_n$ be a sequence of random variables with uniformly bounded $L^1$ norm. Suppose $X_n$ converges in probability to $X \in L^1$.
Is it true that the Cesaro sums $Y_n := \frac{1}{n} \sum_{i = 1}^n ...

-3
votes

0
answers

109
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### Extended dominated convergence theorem in Kallenbergs book [closed]

In the book of Olav Kallenberg Foundations of Modern Probability there is stated a extenden version of the dominated convergence theorem. Its proof and the statement goes exactly:
Theorem 1.23. (...

0
votes

0
answers

42
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### Urn model and recursion

We have an urn with $n$ white balls. In each iteration we pick a ball at random. If it's white, we paint it red and return it to the urn. If it's already red, we discard it. We lose the game if (after ...

9
votes

1
answer

286
views

### Where does the definition of ($\infty$-)groupoid cardinality come from?

The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity
$$\lvert X\rvert := \sum_{[x]...

2
votes

1
answer

104
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### Can we construct close martingales if their terminal marginal laws are close?

Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...

4
votes

1
answer

163
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### Truncated fixed point and regularity structures

This question arose via the helpful comments on this earlier question.
In Hairer's theory of regularity structures, fixed point problems are first solved in certain spaces $D^\gamma$ which consist of ...

8
votes

2
answers

483
views

### On martingale convergence

Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$.
Is it possible that there ...

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votes

0
answers

58
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### $\Gamma$-convergence and ERM problem

I am particularly interested in proving that, when adding Gaussian noise to a dataset $D$, the target functional sequence converges to the target functional without noise as the variance of the noise ...

4
votes

1
answer

221
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### Examples of Borel probability measures on the Schwartz function space?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...

4
votes

2
answers

272
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### Another curious martingale

This is a natural follow up question to A curious martingale.
Does there exist an almost surely continuous martingale that converges in probability to $+\infty$?
Note: We say a process $X_t$ converges ...

0
votes

1
answer

40
views

### Correlation for a Sum of random vectors from the sphere multiplied by matrices

Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...

0
votes

1
answer

50
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### Will the KL divergence between two distributions decrease after passing the fixed channel?

Suppose there are two continuous distributions whose pdfs are $p_1$ and $p_2$, defined on a common support $\mathcal{X}$. Suppose that there is a conditional pdf (the channel) $M:\mathcal{X}\times \...

3
votes

0
answers

115
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### An analogue of Kolmogorov's law of the iterated logarithm

Let $X_1,\dots,X_n$ be independent random variables, each with mean zero and finite variance. Let $S_n = \sum\limits_{k=1}^n X_k$ and $s_n^2=ES_n^2$. We say the sequence obey the law of iterated ...

7
votes

2
answers

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### A curious martingale

Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely?
Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...

3
votes

1
answer

139
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### Continuity of conditional expectation

Let $X$ be a compact metric space, $\mu$ a Borel probability measure on $X$ and $f: X \to \mathbb{R}$ a continuous function. Consider an increasing sequence of $\sigma$-algebras $A_n$ so that for all $...

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votes

0
answers

23
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### Confidence interval of a relative difference [migrated]

I know the size, empirical mean and empirical variance of two samples $X_1$ and $X_2$, but I don't know the values. How can I calculate the bounds of a confidence interval of the relative difference ...

1
vote

1
answer

147
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### A representation formula for the expected value of a stochastic process at a random time

Let $X$ be a continuous stochastic process, and $\tau$ an almost surely positive random variable, not necessarily a stopping time with respect to the natural filtration $\mathcal F_t$ of $X$.
We write ...

0
votes

1
answer

61
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### Existence and uniqueness of a posterior distribution

I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...

0
votes

1
answer

152
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### Does an uncountable convex combination of elements of a set lie in the convex hull of the set in finite dimension?

Suppose that $\mathcal{F}$ is a finite-dimensional vector space and that $C\subseteq\mathcal{F}$ is a convex subset of $\mathcal{F}$.
Is it true that an uncountable convex-combination of elements of $...

0
votes

0
answers

27
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### Tight Chernoff Concentration for Bernoulli(p) RV

I remember seeing a research paper on tight concentration of Bernoulli(p) random variable in terms of $p$.
What I mean is that they used a stronger upper bound for the MGF than $E[e^{s(X-p)
}]\leq e^{...

-1
votes

0
answers

34
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### Sampling vectors and getting close results with high probability

Suppose I have $d$ vectors $v_1,v_2\ldots,v_d$ ($d$ is a very very big number). Each vector has $N$ coordinates. $N$ is not very big, $N\ll d$.
Now proceed in the following way:
Select all 2-element ...

18
votes

2
answers

887
views

### How to show a function converges to 1

Consider the following recurrence relation in two variables:
$$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$
for positive integers $a$ and $b$,
with the boundary conditions $f(0,b)=0$ ...

1
vote

3
answers

226
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### Probability that a 1-D zero mean random walk remains at each step inside a square root boundary

Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\...

4
votes

2
answers

252
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### Paper request : “A random integral and Orlicz spaces” from K. Urbanick

I tried all my methods to find the paper :
“K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, ...

3
votes

2
answers

194
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### Definition of weak conditional convergence of random variables

I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...

2
votes

1
answer

185
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### Expected norm of a product of Gaussian matrices

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?
$$E[\|C_n\|_F^2]=d^{n+1}$$
This ...

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votes

0
answers

43
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### Measurable Function and Inverse Maps [migrated]

Every text I read about random variables starts by introducing the concept of measurable functions. It goes something like this:
Suppose you have 2 measurable spaces $(\Omega, \Gamma)$ and $(\Omega', \...

4
votes

2
answers

317
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### Probabilty measures that are both discrete and continuous

Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...

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votes

1
answer

236
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### On Impossible events

Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...

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votes

0
answers

46
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### The probability to visit a state for the first time after n steps in a markov chain [migrated]

I have the following Markov Chain:
(the probabilities are written above the arrows, and 'a' is a number between 0 and 1)
I want to show that State-1 is a persistent state. To show that, I need to ...

2
votes

1
answer

81
views

### Why do distributional isomorphisms preserve joint distribution?

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and
$$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$
be integrable random ...

0
votes

1
answer

203
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### Question about the proof of Propp-Wilson algorithm in Olle Häggström's book

Update: Oops! This is a stupid question and should be closed. The definition of the probability space that contains events $A_i$ requires using a single random stream.
I have difficulties ...

2
votes

1
answer

90
views

### Thinning of (mixed) binomial point process

Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...

1
vote

1
answer

82
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### Kolmogorov inequality for Bernoulli random variables

This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...

1
vote

0
answers

55
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### What do $\gamma$-radonifying operators radonify?

In the second volume of their Analysis in Banach Spaces, Hytönen et al. introduce the notion of $\gamma$-radonifying operator more or less as follow.
Let $(\gamma_j)_{j\in\mathbf N}$ be a sequence of ...

2
votes

0
answers

76
views

### Random walk with same directions and different step sizes

Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$.
Consider the following two random ...

6
votes

2
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### Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation

$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...

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votes

0
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43
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### Sum of Skellam-distributed number of random variables

Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...

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0
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42
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### Approximating Hamming distance distributions

Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$....

0
votes

1
answer

60
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### Weak convergence to product measure form conditional convergence of marginals

$\newcommand\Ac{\mathcal A}$
$\newcommand\BL{\operatorname{BL}}$
$\newcommand\reals{\mathbb R}$
$\newcommand\eps{\varepsilon}$
$\newcommand\pr{\mathbb P}$
$\newcommand\ex{\mathbb E}$
$\newcommand\...

2
votes

0
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### Construct a Bregman divergence from Wasserstein distance

I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...

2
votes

1
answer

129
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### Some identities from graph theory and probability

The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...

3
votes

0
answers

103
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### Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...

1
vote

0
answers

46
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### functional resembling random variable norm

Let $N\subset\mathbb{R}$ be finite
and define
$$
A(N)
=
\sum_{i
\in\mathbb{Z}
}\min\{
2
^i,
|N\cap[2^i,2^{i+1})|
\},
$$
where
$\mathbb{Z}=\{0,\pm1,\pm2,\ldots\}$
and
$|\cdot|$ denotes set cardinality.
...

3
votes

1
answer

103
views

### Distribution of the change in Hamming distance between two sequences

Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$....

-1
votes

1
answer

65
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### Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?

Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...

4
votes

1
answer

82
views

### Reflecting Brownian motion in disk

What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it?
The transition density ...

7
votes

1
answer

124
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### Nearest neighbors on random complete graph

Consider the complete graph on $2n$ vertices, where the ${2n \choose 2}$ edges have distinct lengths in uniform random
order. So each vertex $v$ has a nearest neighbor $N(v)$, across the shortest ...