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.

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

As no answer or comment to Can we construct close martingales if their terminal marginal laws are close? we consider a simplified version (discrete-time) as below: Let $M=(M_k)_{0\le k\le n}$ be a ...
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Piecewise Ornstein-Uhlenbeck process time integral

Let $X_t$ be a piecewise Ornstein-Uhlenbeck process with infinitesimal variance $\sigma^2$ and (piecewise) infinitesimal mean $\theta_1$ for $x<c$ where $c$ is a constant and $\theta_2$ for $x\geq ...
17miles's user avatar
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1 answer
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Maximize ratio of Chernoff divergence to Bhattacharya divergence

For probability distributions $P, Q$ with pdfs $p, q$, and a parameter $\alpha \in[0,1]$, define $$C_\alpha(P, Q) = -\ln\left(\sum_x p(x)^\alpha q(x)^{1-\alpha}\right).$$ Then define $C(P, Q) = \max_\...
Mark Schultz-Wu's user avatar
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1 answer
70 views

Asymptotic variance for averages of trajectory functionals of Markov chain

I am looking for references on theory for convergence rates of ergodic averages of a Markov chain in the more general setting where the functional is over multiple states or even a whole trajectory, ...
itchidese's user avatar
1 vote
1 answer
109 views

Properties of the relatively bounded probability distributions on the simplex over the natural numbers

Setup : Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...
P. Quinton's user avatar
4 votes
0 answers
808 views

Number of arrangements that contain at least 1 path from top to bottom of 2D matrix

I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white. With that information, I can easily calculate the total number of black element arrangements that exist ...
Cardstdani's user avatar
3 votes
0 answers
129 views

Known relations between mutual information and covering number?

This is a question about statistical learning theory. Consider a hypothesis class $\mathcal{F}$, parameterized by real vectors $w \in \mathbb{R}^p$. Suppose I have a data distribution $D \sim \mu$ and ...
Tanishq Kumar's user avatar
3 votes
0 answers
71 views

Monotone Characteristic Function

Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$. What other properties must $X$ have in ...
Andrea Aveni's user avatar
6 votes
1 answer
352 views

Is a martingale conditioned to be large a submartingale?

Let $X$ be a continuous time martingale such that $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Let $x \in \mathbb R$ be such that $\mathbb P(X_\infty \geq x) > 0$, and define the ...
Nate River's user avatar
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Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?

I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
yohbs's user avatar
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$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
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1 answer
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How to calculate math expectation of recurrent function? [closed]

I try to figure martingale strategy mathematically. Let us say we start with bet $x$ and designate obtained value as $f(x)$ I come up with the following equation $$ \mathbb{E}f(x) = \frac{18}{37} x + \...
Nourless's user avatar
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Why do we define independence for zero-probability events? [closed]

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 ...
Matthias's user avatar
4 votes
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Is the integer factorization into prime numbers normally distributed?

Edit: Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking. Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...
mathoverflowUser's user avatar
3 votes
1 answer
141 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 ...
Nate River's user avatar
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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 ...
leonbloy's user avatar
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10 votes
1 answer
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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]...
Matthew Niemiro's user avatar
2 votes
1 answer
181 views

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 ...
Fawen90's user avatar
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4 votes
1 answer
195 views

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 ...
NZK's user avatar
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9 votes
2 answers
633 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 ...
Iosif Pinelis's user avatar
4 votes
1 answer
261 views

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 ...
Isaac's user avatar
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4 votes
2 answers
327 views

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 ...
Nate River's user avatar
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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 ...
giladude's user avatar
  • 155
0 votes
1 answer
88 views

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 \...
jkfds's user avatar
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3 votes
0 answers
119 views

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 ...
graham's user avatar
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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 ...
Nate River's user avatar
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3 votes
1 answer
169 views

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 $...
A.M.'s user avatar
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1 vote
1 answer
195 views

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 ...
Nate River's user avatar
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0 votes
1 answer
74 views

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 ...
CoilyUlver's user avatar
1 vote
1 answer
175 views

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 $...
rick's user avatar
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0 votes
0 answers
47 views

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^{...
Black Jack 21's user avatar
27 votes
5 answers
3k 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 $f(0,b)=0$ ...
Simd's user avatar
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1 vote
3 answers
246 views

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\...
MathRevenge's user avatar
4 votes
2 answers
260 views

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, ...
Stochastic Student's user avatar
3 votes
2 answers
244 views

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 $\...
Syd Amerikaner's user avatar
2 votes
1 answer
209 views

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 ...
Yaroslav Bulatov's user avatar
4 votes
2 answers
417 views

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 ...
Iris Allevi's user avatar
-2 votes
1 answer
250 views

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)=...
matteogost's user avatar
2 votes
1 answer
86 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 ...
Pavlos Motakis's user avatar
1 vote
1 answer
216 views

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 ...
zemora's user avatar
  • 545
2 votes
1 answer
104 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, \...
mariob6's user avatar
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1 vote
1 answer
119 views

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 $...
Greenhand's user avatar
1 vote
0 answers
64 views

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 ...
P. P. Tuong's user avatar
2 votes
0 answers
89 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 ...
Farzad Aryan's user avatar
7 votes
2 answers
317 views

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://...
TheBestMagician's user avatar
0 votes
0 answers
50 views

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} = -...
Harry L's user avatar
  • 11
0 votes
1 answer
73 views

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\...
passerby51's user avatar
  • 1,639
2 votes
0 answers
66 views

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 \...
John's user avatar
  • 483
2 votes
1 answer
143 views

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, ...
MathMath's user avatar
  • 1,255
3 votes
0 answers
121 views

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 "...
Andrea Marino's user avatar