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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Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
Wenguang Zhao's user avatar
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0 answers
59 views

Concentration of Sample Mode

Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.
Chandramouli's user avatar
2 votes
2 answers
468 views

Probability space with exactly one Brownian motion

Very recently, the following question was asked: Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. ...
Iosif Pinelis's user avatar
3 votes
1 answer
106 views

Asymptotic estimate of the cardinality of $H_{2n}=\{\vec{a} \in [-N,N]^{2n}:\sum_{i=1}^{2n} a_i = 0\}$

Let's suppose $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$. We may then define: \begin{equation} S_n = \sum_{i=1}^n a_i \tag{1} \end{equation} Now, in order to estimate $\lvert ...
Aidan Rocke's user avatar
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Samples from a modified Bernoulli

Given i.i.d samples $X_1, X_2, \cdots$ from Bernoulli($p$) and $1<c<\frac{1}{p}$, is it possible to construct samples from Bernoulli($cp$) under the assumption that $p$ is unknown? If $c\leq1$ ...
Chandramouli's user avatar
1 vote
1 answer
121 views

Reference request concerning order statistics from the uniform distribution

Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\...
Iosif Pinelis's user avatar
1 vote
1 answer
111 views

Conditional distribution/independence

Suppose $X,Y$ and $Z$ are random elements on $(\Omega,\mathcal{A},\mathit{P})$ taking values in the Borel spaces $U,V$ and $V$ respectively. Moreover, let $\mathcal{F}\subset \mathcal{A}$ be a $\sigma-...
Nicolas Bourbaki's user avatar
5 votes
2 answers
1k views

Relationship between $\alpha$-divergences?

I am working with $\alpha$-divergences and was wondering how understand the relationship between the definitions of Renyi and Amari? Renyi: $D_{\alpha}[p||q] = \frac{1}{\alpha - 1} \log \int p^{\...
nico's user avatar
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0 answers
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Wiener isometry for semimartingales

Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying $$ \mathbb{E}\left[ \int_0^{\...
ABIM's user avatar
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4 votes
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On the convergence of the ratio of order statistics of gaps induced by $n$ uniform points on $[0,1].$

In an MO question here @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables ...
kodlu's user avatar
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4 votes
1 answer
689 views

On the largest and smallest spacings for the uniform distribution

Let $Z_1,\dots,Z_n$ be iid random variables (r.v.'s) each uniformly distributed on $[0,1]$. Let $Z_{n:1}\le\cdots\le Z_{n:n}$ be the corresponding order statistics. For $i=1,\dots,n-1$, let $G_i:=Z_{n:...
Iosif Pinelis's user avatar
3 votes
1 answer
300 views

Expected value of "longest bit / shortest bit" in $n$ uniformly distributed points on $[0,1]$

Let $n\geq 2$ be an integer. We pick $n$ points in $[0,1]$ with uniform distribution. Let $A$ be the minimum distance that two adjacent points have, and let $B$ be the maximum distance that two ...
Dominic van der Zypen's user avatar
213 votes
0 answers
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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
Sil's user avatar
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Finding E[X] based on infinitely many values of its moment generating function

Suppose X is a random variable. I have infinitely many values of the form $E[e^{k \alpha X}]$ for $k=0,1,2,3,...$ where $\alpha \in (0,1)$ is a constant real number. Based on these values is there any ...
user144888's user avatar
15 votes
2 answers
671 views

On sums of independent random variables in Banach spaces

Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that $$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any ...
Lviv Scottish Book's user avatar
0 votes
0 answers
74 views

Optimizer of a semi-discrete optimal transport problem

Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user avatar
7 votes
0 answers
563 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
Tom Copeland's user avatar
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6 votes
1 answer
215 views

De Finetti-style theorem for Point Processes

I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies: Complete independence (the random variables $\eta(B_1), \ldots, \eta(B_n)$...
Noah Stein's user avatar
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2 votes
0 answers
75 views

$\sigma$-fields as closure systems

Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is ...
passerby51's user avatar
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2 votes
1 answer
358 views

Existence and uniqueness of a stationary measure

This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure. Recently I have posted the following question on MO ...
Matheus Manzatto's user avatar
2 votes
0 answers
81 views

Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?

$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
Turbo's user avatar
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2 votes
1 answer
226 views

Overview of interpretations of classical probability

The Stanford Encyclopedia of Philosophy has a nice overview of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography. What books would ...
Tom Copeland's user avatar
  • 9,937
2 votes
2 answers
246 views

Model for random graphs where clique number remains bounded

In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?
Nicolas Boerger's user avatar
1 vote
0 answers
68 views

Concentration or distribution of the scaled $l_p$ norm of a correlation matrix

Background: Among Hermitan random matrices, correlation matrix has a lot of applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the ...
Chee's user avatar
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1 vote
0 answers
216 views

Gaussian mean width of normal random cones

Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by $$ w(T) := \mathbb E \sup_{x \in T} \...
bashfuloctopus's user avatar
2 votes
0 answers
113 views

Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution

Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions. One can ...
R Hahn's user avatar
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4 votes
1 answer
401 views

Concentration inequality for the law of iterated logarithm

The following question arose in one of my research projects. Before stating it, let me give a short background. We all know the law of iterated logarithm. It states that if $X_1,\ldots,X_n$ are i.i.d. ...
Somabha's user avatar
  • 123
1 vote
1 answer
150 views

Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...
Tom's user avatar
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3 votes
0 answers
176 views

Theoretical framework for a divergent random series

Consider the following random variables The $\{m_n\}_{n\geq 1}$ are iid and satisfy $$\mathbb{P}(m_{n}\leq x)\leq C x$$ for $x>0$ and some $C>0$. The $\{L_{n,m}\}_{m\geq n\geq 1}$ satisfy $L_{n,...
Thomas Kojar's user avatar
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5 votes
1 answer
204 views

Conditional expectation of random vectors

$\newcommand{\E}{\mathsf{E}}$ $\newcommand{\P}{\mathsf{P}}$ The following additional question was asked in a comment by user Oleg: Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ ...
Iosif Pinelis's user avatar
5 votes
1 answer
364 views

Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say $$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$ Now if ...
Sascha's user avatar
  • 506
6 votes
2 answers
429 views

A property about probability distribution

Suppose $g(x)$ is a pdf function and k is a positive real number. Let $F(\alpha)=\int_{-\infty}^{\infty}\frac{1}{\frac{g(x+\alpha)}{g(x)}+k}g(x)dx$, where $\alpha$ is positive. I feel $F(\alpha)$ is ...
Peter's user avatar
  • 177
2 votes
1 answer
110 views

Sufficient conditions for inequality with integral of reliability functions

Let $Y$ and $W$ be two random variables with support $(y_1,y_2)$ and $(w_1,w_2)$ and distributions $F_Y$ and $F_W$, both twice continuously differentiable (densities $f_Y$ and $f_W$). Assume that both ...
Ararat's user avatar
  • 143
4 votes
2 answers
363 views

Basic properties of expectation in non-separable Banach spaces

$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear ...
Oleg's user avatar
  • 911
1 vote
0 answers
145 views

Rate of convergence of empirical distribution with respect to Wasserstein distance induced by binary cost function

Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\...
dohmatob's user avatar
  • 6,716
1 vote
0 answers
89 views

Probability space with countable subset such that every subset of positive measure meets the subset

Let $(X, \mathcal F, P)$ be a probability space. Question What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that $\forall$ measurable $A \subseteq X$, $P(A) >...
dohmatob's user avatar
  • 6,716
2 votes
0 answers
183 views

Girsanov density as a functional on $C[0,1]$

I'll formulate the question via an example. On $( C[0,1], \mathcal{C} )$, where $C[0,1]$ is the set of continuous functions on $[0,1]$ and $\mathcal{C}$ the Borel $\sigma$-algebra given by uniform ...
Michael's user avatar
  • 263
3 votes
1 answer
151 views

Central limit type theorems for compact Hausdorff topological groups?

Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure $(\tau*\mu)(A)=\int\int1_A(xy)d\tau(x)...
Jess Boling's user avatar
1 vote
1 answer
585 views

Probability that random Bernoulli matrix is full rank

This is probably known already, but I could not find a quick argument. Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
hookah's user avatar
  • 1,096
1 vote
0 answers
296 views

Wasserstein distance between rotated conditional distributions

Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \...
Terzo's user avatar
  • 11
0 votes
1 answer
155 views

Marginal probability mass function

I have the joint PMF $\exp(y_1 \ln(\lambda)+y_2 \ln(c)+y_2\ln(\lambda)-\ln(y_1!y_2!)-\lambda(1+c))$ for a constant $c>0$. In canonical representation and mixed parameterization I have $\mathbf{\...
Orongo's user avatar
  • 111
1 vote
1 answer
266 views

Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)?

Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x_i$ generated by RAND() function of Excel). I ...
Murali's user avatar
  • 51
7 votes
1 answer
458 views

A theorem by Harald Cramér?

In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement: Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...
Chain Markov's user avatar
  • 2,618
3 votes
1 answer
353 views

Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
Matheus Manzatto's user avatar
5 votes
1 answer
325 views

Bounding the sensitivity of a posterior mean to changes in a single data point

There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...
Ben Golub's user avatar
  • 1,058
2 votes
0 answers
64 views

Convergence of gPC expansions for random variables in the total variation distance

Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
user avatar
4 votes
1 answer
406 views

Stochastic processes and continuity of expectation

Let $X$ be a stochastic process with a.s. continuous sample paths on $[0, 1]$ such that $\mathbb E [X_t]$ is finite for all $t \in [0, 1]$. Given any non null subset $Y$ of the probability space, ...
James Baxter's user avatar
  • 2,029
2 votes
1 answer
97 views

"Сross сubic variation" of two Brownian motions and interpretation of the simulation result

Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$. How to calculate the expression below? Can we rewrite ...
Stephen Paul's user avatar
4 votes
0 answers
184 views

Probability that a Random Monic Polynomial Has Few Real Zeros

In the paper https://arxiv.org/pdf/math/0006113.pdf, it is shown that the probability that a random polynomial $a_0 + a_1x + \cdots + a_n x^n$ has $o(\log n / \log\log n)$ real zeros is $n^{-b + o(1)}$...
Ashvin Swaminathan's user avatar
1 vote
2 answers
232 views

Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings

Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means. Question. Given $\alpha > 0$, what is value of, ...
dohmatob's user avatar
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