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.
9,371 questions
0
votes
1
answer
37
views
Limit uniform distribution of moving particle in simple polygon with random reflections
Let $G, \partial G$ be the interior and the boundary of simple polygon respectievly. For $p \in \partial G$ let
$$
D_p=\left\{ d \in S^1: \exists\varepsilon > 0 \text{ s.t. } p+\delta d \in G, 0 &...
- 496
1
vote
0
answers
35
views
Recovering the additive channel model from sum power constraint
The Additive White Gaussian Model ($\mathsf{AWGN}$) model is the following: You send a message $x$ from a finite set of real alphabets $\chi$ and White Gaussian Noise (noise of Gaussian distribution $\...
- 73
-2
votes
0
answers
93
views
Financial Interpretation of the Lebesgue-Stieltjes Integral
I asked this question in the Quantitative Finance stack exchange (https://quant.stackexchange.com/questions/85294/financial-interpretation-of-the-lebesgue-stieltjes-stochastic-integral) and it was ...
2
votes
0
answers
114
views
+100
Is there a name for this type of probabilistic predictability of stopping times?
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space, and let $\tau \colon \Omega \to [0,\infty]$ be an $(\mathcal{F}_t)$-stopping time.
We will say ...
- 828
3
votes
1
answer
261
views
Do we have a version of Hoeffding's inequality for these non-independent variables
I am writing a probabilistic argument (and I am not a probability theory expert), and the following would be useful to me. I tried asking AI but the answers did not seem helpful, so hopefully this is ...
- 13.1k
5
votes
0
answers
190
views
Stochastic dominance $ \tanh Y\tanh Z \succeq \tanh X $
Let $X,Y,Z$ independent Gaussian r.v.'s with mean=variance. Let's denote these mean/variance parameters by $g_X,g_Y,g_Z>0$ respectively.
Set $T_1:=\tanh X$, $T_2:=\tanh Y\tanh Z$.
My question. ...
- 363
6
votes
1
answer
507
views
Conditional probability is the same on all coin flips . Does it imply independence?
Consider a probability space $(\Omega,\mathcal{F},P)$.
Consider an infinite sequence $(B_i)$ of mutually independent events with the same probability $p$ (coin flips). Write $S$ the $\sigma$-algebra ...
- 321
1
vote
1
answer
199
views
Conditional probability is the same on all events with probability $p$. Does it imply independence?
Consider a probability space $(\Omega,\mathcal{F},P)$
Consider a divisible $\sigma$-algebra $\mathcal{S}\subset\mathcal{F}$, an event $A\in\mathcal{F}$ and a number $p\in (0;1)$. Assume the ...
- 321
2
votes
1
answer
125
views
Convergence of the correlation matrix for a specific type of random matrix product
Let $X= BM$ where
$B$ is a random $n\times m$ matrix with independent elements uniformly distributed on $[a, b]$.
$M$ is random $m\times m$ matrix with independent elements uniformly distributed on $[...
- 3,180
-4
votes
0
answers
128
views
Most probable path between two points, Onsager–Machlup functional
Say I have an SDE $$dx_t = b(x_t)dt + dB_t$$ where $B_t$ is a Brownian motion. I would like to compute the most probable path between two points $y_0$ and $y_1$.
I know the Onsager–Machlup functional ...
- 41
2
votes
0
answers
111
views
Optimal constant in $L^1-L^2$ inequality on Gauss space
For a differentiable real-valued function on $\mathbb{R}^n$, denoting $\partial_i f$ for the $i$th partial derivative, we can define the functional
$$
T_n(f) = \sum_{i=1}^n \frac{1}{1 + \log(\|\...
- 582
4
votes
1
answer
172
views
Explicit examples separating Wasserstein distances from Jensen–Shannon divergence?
I am looking for concrete, preferably elementary, examples of pairs of probability measures $(\mu_n,\nu_n)$ on a common metric space (e.g. $\mathbb{R}^d$) that explicitly demonstrate the non-...
- 41
1
vote
0
answers
27
views
Probability of Generating an Ideal by Two Bounded-Coefficient Random Combinations of Given Generators
Let $K = \mathbb{Q}(\zeta_{p^k})$ be a prime-power cyclotomic field with ring of integers $\mathcal{O}_K$. Fix a $\mathbb{Z}$-basis $\underline{\omega} = (\omega_1, \dots, \omega_d)$ of $\mathcal{O}_K$...
- 11
-1
votes
0
answers
120
views
Time-indexed probability measures
I will try to demonstrate the problem with a trivial example:
Suppose that there is a traffic light with the usual three distinct colors: Red, White and Green. The traffic light will switch on one ...
- 155
5
votes
1
answer
269
views
Stochastic order of generalized chi-square distributions
For any $a\in \mathbb{R}^n_{>0}$ define the random variable $\chi^2_a=\sum_{i=1}^n a_i Z_i^2
$
where the $Z_i\sim \mathcal{N}(0,1)$ are i.i.d. Given $a,b\in \mathbb{R}^n_{>0}$ I am looking for ...
- 105
1
vote
0
answers
38
views
Justifying the Robbins-Monro procedure using Dvoretzky's theorem on stochastic approximation
A colleague and I are trying to understand some results in stochastic approximation theory with a view to gaining quantitative information about rates of convergence of certain processes. We have done ...
- 1,169
1
vote
1
answer
105
views
Does uniform strict convexity of a local lattice action imply a uniform Brascamp–Lieb inequality?
Consider a sequence of finite-dimensional probability measures $\mu_n$ on $\mathbb{R}^{d_n}$ given by$$\mu_n(dx) = Z_n^{-1} e^{-S_n(x)}\,dx,$$where $x \in \mathbb{R}^{d_n}$, and $Z_n$ is the finite ...
- 233
1
vote
1
answer
319
views
What proportion of vectors in ${\mathbb{F}_2^n}$ have more than $\frac{n+\sqrt{n}}{2}$ ones
Ben Green's "Finite field models in additive combinatorics" (proof of theorem 9.4) states that for sufficiently large $n$, the set $A$ of vectors in ${\mathbb{F}_2^n}$ with more than $\frac{...
16
votes
2
answers
1k
views
Expected number of upsets in a knockout tournament
$2^n$ players $P_1, \dots, P_{2^n}$, ordered in decreasing order of skill are placed uniformly at random at the leaves of a binary tree of depth $n$.
They play a knockout tournament according to the ...
- 9,980
0
votes
0
answers
39
views
Limit of expectation of a Markov process at stopping times
Suppose $X \in \mathbb R^d$ is a Markov process with cadlag paths and a stationary distribution $\mu$. Let $\tau_n : =\inf\{t\ge 0: |X(t) | \ge n\}$ and assume that $X$ is non-explosive and hence $\...
- 622
2
votes
1
answer
161
views
Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
- 818
0
votes
1
answer
135
views
Measurability of $t \mapsto \int_A f(t, \omega)\mathbb{Q}_t(\mathrm{d}\omega)$ when $(t, \omega) \mapsto f(t, \omega)$ is not measurable in $t$
I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...
- 208
1
vote
0
answers
40
views
Speed measure for sticky drifted Brownian motion
Consider the following SDE:
$dD_t=1_{\{D_t>0\}}d(B_t+\sqrt{2}t)+dL_t\\dL_t=\sqrt{2}1_{\{D_t=0\}}dt$
under the constrain that $D_t\geq 0$, where $L_t$ is the local time of $D_t$ defined by Tanaka's ...
- 41
5
votes
1
answer
182
views
Integral representation of Markov operators
On a measurable space $(E,\mathcal E)$, a stochastic kernel is a function $p\colon E\times \mathcal E\to [0,1]$ such that:
for each $x\in E$, the function $A\mapsto p(x,A)$ is a probability measure;
...
- 286
0
votes
0
answers
71
views
Cumulants of random inner product on the sphere
Let $Z={U}^\top {V}$ where ${U}$ and ${V}$ are uniformly distributed $\mathbb{S}^{p-1}$. It is known that $Z$ has even moments given by
\begin{align*}
\mathbb{E} Z^{2m} = \frac{(2m-1)!!}{p(p+2)\cdots(...
- 29
1
vote
0
answers
94
views
solve explicitly an integral equation
Let $\lambda$ be the Lebesgue measure. Let $f \in L_1([0,1])$, I would like to construct a $g$ function in $L_1(\mathbb{R}^+)$ such that
$$
\mathbf{1}_{[0,1]}\lambda(dx)\text{-a.e., }\quad f(x)
= \...
- 696
2
votes
0
answers
78
views
Martingale central limit theorem: simple version reference
I don't know if it is better to ask here or on MSE, if that's the case I can post the question there. I would need a simple version of the martingale central limit theorem. And, by simple, I mean the ...
- 911
0
votes
0
answers
90
views
Is the Matérn field $(\kappa^2 - \Delta)^{\alpha/2} u = W$ the stationary distribution of an infinite-dimensional Ornstein-Uhlenbeck SDE?
In spatial statistics, the Matérn Gaussian field on $\mathbb{R}^d$ is often defined as the (weak) solution to the SPDE
$$
(\kappa^2 - \Delta)^{\alpha/2} u \;=\; W,
$$
where $W$ is Gaussian spatial ...
- 787
2
votes
0
answers
95
views
Inverting the conditional expectation for some coupling
Let $\nu$ be a probability measure equivalent to $\mathbf{1}_{\mathbb{R}_+}(y) \, \lambda(dy)$. Let $\pi$ be a probability measure on $\mathbb{R}^2$ of second marginal $\nu$, such that $\nu(dy)$-a.e.,
...
- 696
0
votes
0
answers
18
views
Solving decoupleable families of FBSDEs
Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs
\begin{align}
Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
- 4,842
2
votes
1
answer
128
views
How to prove the convergence of the maximum point random variable of random concave function sequence？
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
- 35
1
vote
1
answer
182
views
A question related to the convergence of a sequence of random variables
Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables adapted to the filtration
$\{\mathcal{F}_n\}_{n \geq 1}$. If $X_n \to X$ a.s., does it follow that
\begin{equation*}
\mathbb{E}\left[X_n \, \...
3
votes
1
answer
426
views
References for this law?
Let $S$ be a symmetric simple random walk starting at $0$ and denote by $p_{n,k}$ the probability $S$ occupes $k\in\mathbf{Z}$ at time $n\in\mathbf{N}$. For any $n\in\mathbf{N}$ denote also $q_n=\left(...
- 81
0
votes
0
answers
103
views
Monotonicity of the convex sum of two binary entropy functions
Let $T$ be some random variable on $[0,1]$, and define
\begin{equation}
\alpha(t) \triangleq \mathbb{E}[T \vert T\le t],\\
\beta(t) \triangleq \mathbb{E}[T \vert T>t], ~t\in[0,1].
\end{equation}
...
- 570
2
votes
0
answers
86
views
The Nash equilibrium of an attack-defense game with infinite energy levels
It is not hard to find the Nash-equilibrium of an attack-defense game with finite energy levels by solving a finite number of equations, but how to solve an infinite number of equations?
In this game, ...
- 83
1
vote
1
answer
148
views
Distance between two Gaussian measures on a Hilbert space
$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\...
- 961
2
votes
2
answers
360
views
Worst-case number of single-card insertions to force Player 1 to be the unique winner in Texas Hold’em
Recently, I have been wondering about how to stack a deck in my favor using minimal moves for Poker. Concretely, I want to know if any deck can be stacked in my favor in 2 or 3 card moves. I have been ...
- 31
1
vote
0
answers
88
views
Weak convergence of nets of measures in a locally convex space
Let $X$ be a locally convex space, let $(p_t)_{t\in T}$ be a net of Borel probability measures on $(X,\sigma(X,X^*))$, and let $p$ be a $\tau$-additive (in particular, Radon) with respect to the weak ...
- 341
6
votes
0
answers
241
views
QR code problem
Motivation. Today, I saw a QR code with an unusually large black square (a largish group of “pixels” coloured black and forming a square). This inspired the following problem.
Problem. Fix $n\in\...
- 54.4k
4
votes
0
answers
144
views
Are random polytopes expanders?
Let $P$ be a random polytope defined by the intersection of $N$ random halfspaces $H_i$ in $\mathbb{R}^n$ (here I define a random halfspace as $x \cdot X_i \leq 1$ for a random vector $X_i$, say ...
- 401
0
votes
1
answer
74
views
Convex concentration of a tensor-squared spherical vector
Let $X$ be a random vector in $\mathbb{R}^n$. We say that $X$ has the convex concentration property with constant $K > 0$ if for every $1$-Lipschitz convex function $\varphi : \mathbb{R}^n \to \...
- 39
2
votes
2
answers
236
views
Is there a fully one-sided version of Etemadi's inequality?
$\def\P#1{\mathbf{P}\!\left[#1\right]}\def\E#1{\mathbf{E}\!\left[#1\right]}$Let $X_1, \dots, X_n$ be independent real random variables (but not necessarily i.i.d.), and let $S_t = \sum_{i = 1}^t X_i$ (...
- 740
0
votes
1
answer
129
views
Uniform integrability for nets of measures under weak convergence
Let $X$ be a topological space (or metric space if needed) and let $(p_t)_{t\in T}$ be a net of Borel probability measures on $X$ which converges weakly to a Borel probability measure $p$, that is, ...
- 341
4
votes
0
answers
204
views
Uniformly picking numbers without memory [duplicate]
Motivation. For my older son's spelling bee contest, he gave me a list of $n$ difficult words to read to him, so he could write them down. As I was filling the dishwasher at the same time, I didn't ...
- 54.4k
0
votes
0
answers
37
views
Characterizing exponential families with elementary normalizing transformations
Let $f_\theta(x) = \exp(\theta T(x) - K(\theta))$ be a one-parameter exponential family of probability density functions with respect to the Lebesgue measure on $\mathbb{R}$, for $\theta$ in an open ...
- 763
15
votes
2
answers
1k
views
Can Lebesgue's differentiation theorem fail almost everywhere?
Let $(X,d,\mu)$ be a metric measure space. Does there exist $f\in L^1(X,\mathbb R)$ so that
$$\mu\left(\left\{x\in X:\lim_{r\to 0^+}\frac{\int_{B_r(x)}f(y)d\mu(y)}{\int_{B_r(x)}d\mu(y)}= f(x)\right\}\...
- 2,345
0
votes
0
answers
78
views
Joint density of $(B_T^2, \int_t^T B_s^2 \, ds)$ for 1D Brownian motion
Let $B = (B_s)_{s \in [0,T]}$ be a standard one-dimensional Brownian motion. Define the two random variables: $U = B_T^2$
and $V = \int_t^T B_s^2 \, ds$, for some fixed $ t \in [0,T)$.
I am ...
- 422
0
votes
0
answers
59
views
How does uniform weak convergence carry over to evaluating an empirical process at an estimated parameter?
Let $ \mathcal{F} = \{ f_\theta : \theta \in \Theta \} $ be a class of functions indexed by $ \theta $. If $ \mathcal{F} $ is a Donsker class, then the empirical process $ \mathbb{G}_n(f_\theta) = \...
- 55
3
votes
1
answer
269
views
Probability measures on a non-separable space
I am reading Chapter 15 of Aliprantis & Border's "Infinite Dimensional Analysis: A Hitchhiker's Guide". Specifically, we start with a metrizable space $X$ and the space of probability ...
- 217
2
votes
0
answers
192
views
Maximum vs sum of squares: role of covariance
I want to do inference on a random vector $X = (X_1, X_2, \dots, X_n)$ with covariance $\Omega$. For that purpose, I consider the maximum statistic $\Vert X\Vert_\infty = \max_{i=1}^n\vert X_i\vert$ ...
- 49