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.
920 questions
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Does independence of the sequence $f(A_i, B)$ imply the sequence is independent of $B$?
Suppose $B, \{A_i: i \in \omega\}$ are i.i.d. random variables with uniform distributions on $[0,1]$. If $f$ is a map such that $\{f(A_i, B): i \in \omega\}$ are independent, must $\{f(A_i, B): i \in \...
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is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
- 521
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Does convergence in law to absolutely continuous limit imply convergence in convex distance?
Let $(X_n)$ be a sequence of $\mathbb{R}^d$-valued random variables converging in distribution to some limiting random variable $X$ whose CDF is absolutely continuous with respect to the Lebesgue ...
- 617
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Intuition behind Gubinelli derivative
I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject.
On page 14 of "A Course on Rough Paths
With an ...
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Mathematics research relating to machine learning
What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
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424
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Existence of an invariant measure on an infinite dimensional space via Lyapunov functional
Set-up.
Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V ...
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Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle
What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The answer appears to be uniform measure, since informally it appears better to have ...
- 3,209
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Fourier transform of a simple random walk
Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there ...
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Can a Brownian motion be fast at its extrema?
After pondering this MO question > Location of maximum of Brownian motion with rough drift <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its ...
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Full-rank rectangular matrices over GF(2)
Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
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287
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Infinite tridiagonal matrices and a special class of totally positive sequences
Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix
\begin{equation}
T(\Bbb{y}) := \,
\...
- 2,137
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Moment Bounds on Hölder norms of stochastic processes
It is relatively easy to show that a stochastic process is Hölder continuous using Kolmogorov continuity theorem link text. But how does one obtain a bound $\mathbb{E} \left\Vert u\right\Vert _{\gamma}...
- 1,256
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Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
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Randomized version of Turán's theorem
Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...
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Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
- 503
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266
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Throwing darts at a barn and putting a bullseye around them in higher dimensions
Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice)
Let $B$ be a ball of radius $1$.
Consider the ratio
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
- 148k
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A polynomial identity involving Wick ordering of a complex power
The problem is related to the paper 1509.02093 by Oh and Thomann, where the authors considered the 2D Wick ordered NLS.
Let $g=a+ib$ be a complex number. Then it is claimed (see (2.7) in the paper and ...
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The expectation of two sides of rectangle is equal. Can we deduce that in the expectation the rectangle is not very far from being a square?
Let $T$ be a set of $n\ge 3$ points in the plane such that not all of them lie in a common line. Pick two distinct points $\{a=\left( \begin{array}{c} a_{1} \\a_{2} \end{array} \right) ,b=\left( \...
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Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
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Which coupling of uniform random variables maximises the essential infimum of the sum?
Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$.
Question: Let $\mu_1, \dots, \mu_n$ ...
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Is this process strictly positive?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and $\...
- 1,399
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Does central limit theorem hold for general weakly dependent variables?
Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of ...
- 4,466
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625
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How to find the "natural scale function" for more general stochastic processes?
In stochastic analysis, for an Ito diffusion $X_t$ such that $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$, we can exlpicitly compute a "natural scale function"
$$S(x)=\int^x\exp\left(-\int^y\frac{2\mu(...
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Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
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Is there an inequality relation between KL-divergence and $L_2$ norm?
According to the Pinsker inequality, we have the following inequality:
\begin{equation}
\delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q),
\end{equation}
where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}...
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980
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"Nice" eigenvectors for (square of) adjacency matrix of a bipartite graph?
Let $G$ be a bipartite graph, and let $A$ be its adjacency matrix.
I was wondering in this case whether $A^2$ will have nice eigenvectors that reflect combinatorial structure of the graph. I'd be ...
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Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
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When is the limit of Martingales a Martingale?
I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$. Suppose that there is a filtration $F_t$ such that for each $n$, $X_n$ is a martingale with respect to this ...
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Optimisation of betting strategy
Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we can'...
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Limit of a sum with binomial coefficients
Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$
$$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$
$$C_k = \frac{\sum_{i=1}^k(...
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Involutions on $[0,1]$ given by power series (related to probability generating functions)
Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$.
Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and $a_k\...
- 3,937
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Reference request: Urbanik's work on random integrals and Orlicz spaces
Several important papers on Lévy processes are referring to the following paper:
K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces,
Bulletin de l'Académie Polonaise des Sciences, ...
- 2,306
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One can earn nothing on the Brownian motion, true ?
Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$).
Consider some "trading strategy" ...
- 24.9k
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Spherical average of $\frac{1}{x}$
Let $X_1,...,X_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$
Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
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Existence of a joint distribution on Bernoulli variables with same probability of being pairwise different
Let $m\in\mathbb{N}$ and $p\in(0,1)$ be arbitrary. Is there a sequence $X_1,\dots,X_m$ of random variables with the following specs on their distribution:
Each $X_i$ is unbiased Bernoulli: $X_i\sim {\...
- 1,096
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625
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Moments of the Hölder norm of Brownian process
It is well known that for a brownian process $B(t),t\geq 0$, it holds
$$
\sup_{0\leq s<t\leq T}\frac{|B(t)-B(s)|}{|t-s|^\alpha}<\infty
$$
almost surely, for any $T>0$ and $\alpha<1/2$.
...
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inequality in a shape of inclusion exclusion formula
I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers $a_1,a_2,...
- 341
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2
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193
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Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
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Scaling of First-passage times for Random Walk on integer lattices
Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...
- 61
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665
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Concentration of sums of random vectors under moment conditions on the marginals
Let $X_1,\dots,X_n\in {\bf R}^d$ be $d$-dimensional iid zero mean random vectors with covariance matrix $\Sigma$. I am interested in tail bounds for the Euclidean norm
$$N_n\equiv \frac{1}{\sqrt{n}}\|...
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Estimate probability( 0 is in the convex hull of N random points ) ?
Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour ...
- 265
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Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 393
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139
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What is the expected value of the submeasure of a random set?
Let $N \in \mathbb N$ and suppose that $\phi$ is a submeasure on $[1,N] = \{1,2,\dots,N\}$, by which I mean that $\phi$ is a function $\mathcal P ([1,N]) \rightarrow \mathbb R$ such that
i. $A \...
- 18.6k
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Bounding Entropy in terms of KL-Divergence
Let $h(X)$ be the differential entropy of a continuous random variable $X$ with density $f$, and let $Y$ be another continuous random variable with density $g$. If $KL(X\mid\mid Y)$ is the Kullback-...
- 49
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182
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Determine the minimal elements of a Dynkin system generated by a finite set of finite sets
(This is a refined version of https://cs.stackexchange.com/q/144371)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
- 5,822
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124
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The behavior of a uniform order statistic near zero
Let $X_{(k)}$ be the $k$th order statistic out of $n$ uniform $[0,1]$ random variables. Let $q$ be the location of the $p$ quantile of $X_{(k)}$, i.e. $\Pr[X_{(k)}\leq q] = p$. For small $p$, Is it ...
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3
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1k
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Incremental entropy computation
After a quick internet search I found no method for incremental entropy computation.
Question 1
Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...
- 161
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1
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365
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Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
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830
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Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
- 141
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225
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Is the inequality of the random matrices correct?
I am not familiar with random matrices but I need to confirm the correctness of the inequality below.
Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let
$A_1,\ldots, A_n$ be $m\times m$ ...
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