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.
9,022 questions
0
votes
0
answers
81
views
White noise: a tempered distribution version of the stochastic convolution
Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...
- 573
0
votes
0
answers
67
views
"Marginal" Three-Series Theorem?
Let $\{X_t\}_{t=1}^\infty$ be a sequence of random variables and define the partial sums by $S_t=\sum_{s=1}^t X_s$.
Kolmogorov's three-series theorem states that
Theorem (Davidson, 2021, Theorem 21.8)...
- 73
0
votes
0
answers
73
views
Criteria giving sufficient conditions for a Borel measure to have compact support
I am interested in criteria that guarantee that a Borel probability measure has compact support.
I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient ...
- 870
0
votes
0
answers
54
views
Reference request: "doubly empirical" measure associated to a random measure
I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
- 660
0
votes
0
answers
35
views
Question about the spectrum of a deformed GOE matrix
Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the ...
- 1
0
votes
0
answers
87
views
Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
0
votes
1
answer
68
views
Convergence of edge eigenvalues for Gaussian matrices
I am reading this lecture note.
I have a difficulty in understanding the third section in chapter 6. Particularly, in Theorem 4.1, they claimed that
Let $X$ be a Gaussian Wigner matrix satisfying ...
- 129
0
votes
0
answers
41
views
Formalization of sample convergence
Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that
$$X_i = \sigma_i z_i,$$
where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
- 33
0
votes
0
answers
22
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
0
votes
0
answers
76
views
Convergence of probabilities imply convergence of joint probability
Context: Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\...
- 33
0
votes
0
answers
42
views
Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
0
votes
0
answers
90
views
Martingale defined by an integral
Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
- 573
0
votes
0
answers
73
views
Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
0
votes
1
answer
92
views
Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
- 215
0
votes
0
answers
91
views
Some new questions on Rademacher complexity
For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
- 185
0
votes
0
answers
99
views
Random walks on groups
I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
- 9
0
votes
0
answers
114
views
Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
0
votes
0
answers
82
views
Conditional distributions of random orthogonal projection matrix
I have encountered a rather curious question.
Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...
- 319
0
votes
0
answers
21
views
Proof that Component-wise MH algorithm is invariant w.r.t. target measure
consider a standard situation in Bayesian modelling,
given real vector parameter $\theta=(\theta_1,\dotsc,\theta_n)$ and observations $x$ we derive a posterior distribution $\pi$ with posterior ...
- 31
0
votes
0
answers
71
views
Weak limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment
Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that the pushforward $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges ...
- 69
0
votes
0
answers
39
views
Multivariable local CLT for uncorrelated (but dependent) coordinates?
Let $\vec f, \vec g\sim\mathcal{N}(0, \sigma^2I_n)$ be independent Gaussians.
Define $\mathsf{cyc}^i(\vec f) = (\vec f_i, \vec f_{i+1},\dots, \vec f_{n-1}, \vec f_0, \vec f_1,\dots, \vec f_{i-1})$ to ...
- 2,183
0
votes
0
answers
82
views
High probability bound on number of sparse solutions to Gaussian linear system
Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...
- 43
0
votes
0
answers
19
views
Analyzing point distributions in Voronoi tessellations from two probability sources $p$ and $\tilde{p}$
Let's suppose I have a probability distribution $p$ and another distribution $\tilde{p}$. Suppose I sample $K$ points from the distribution $p$ which will be my centroids for my Voronoi tessellation. ...
- 101
0
votes
0
answers
43
views
For a convex body $K\subset R^n$, does the quantity $\min_{t>0} E[t^{-1} \|tg - \pi_K( t g)\|_2]$ have a name? Where has it been studied?
Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).
Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as
$$
\Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\|
$$...
- 1,724
0
votes
0
answers
49
views
Weak convergence of Gibbs measures with converging energy functions
Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$
and let $Q$ be a fixed probability measure on $A$.
For each $\theta>0$, define the probability ...
- 503
0
votes
0
answers
50
views
Approximate CDF of integral using the Berry-Esseen theorem
I'm trying to approximate CDF of the integral $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is the Wiener process, i.e. $W_t\sim N(0,t)$.
For this I use ...
- 49
0
votes
0
answers
77
views
Texts on coalescent theory/probability methods for DNA evolution
I am starting a PhD on mitochondrial evolution modelling with a focus on probabilistic methods and coalescent theory. For this purpose, I am looking for
advanced textbooks on probability methods for ...
- 203
0
votes
0
answers
116
views
Concentration bounds for sum of weighted sampling without replacement
Let $X$ be a collection of $2l$ non-negative numbers $X_1,X_2,\ldots,X_{2l}$. We draw $l$ weighted (proportional to values) samples without replacement from $X$. Let $S$ denote this set of $l$ samples....
- 11
0
votes
0
answers
55
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} = -...
- 11
0
votes
0
answers
142
views
Calculating the expected hitting time of a specific birth and death chain
I am working with a specific birth and death chain, defined as follows.
Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
0
votes
0
answers
101
views
Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 620
0
votes
0
answers
73
views
Computationally efficient solution for the measure of central tendency minimizing Lp loss for p > 1
We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$
For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of ...
- 109
0
votes
0
answers
55
views
Modeling player interactions in multi-dimensional rating systems
In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the ...
- 1
0
votes
0
answers
107
views
How to show that the map $ R $ here is measure-preserving
Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...
- 1,219
0
votes
0
answers
140
views
Bounding the difference of weighted-mean estimations
I recently asked a question related to a proof I'm working on as well as a follow-up to that. The detailed counter-examples I received elucidated a lot, but it quickly transpired (i) that I was ...
- 15
0
votes
0
answers
25
views
Maximum sizes of independent sets in (non-uniform) hypergraphs
It is a very well understood problem to compute the size of the maximum independent set in a uniform hypergraph (in terms of extra conditions).
My question is the following: what is known for ...
- 1,561
0
votes
1
answer
450
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 13
0
votes
0
answers
161
views
Markov process with time varying transition kernels
I cross post this question from StackExchange as it may be more appropriate.
I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
0
votes
0
answers
87
views
coupling method for first hitting times
Consider a Markov process $(X_t: S \to S)_{t \ge 0}$ that begins with two initial probabilities $\mu_1$ and $\mu_2$ defined on the state space $S$. Let's define the first hitting time $\tau$ as $\tau:=...
user513728
0
votes
0
answers
95
views
Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
0
votes
0
answers
230
views
Lower bounds for sub-Gaussians?
For a random variable $X$, define
$$\lVert X\rVert_{\psi_2} =\inf \{k>0\mid \mathbb{E}[\exp((X/k)^2)]\leq 2\}$$
and for a random vector $\vec X$, define
$$\lVert \vec X\rVert_{\psi_2} = \sup_{\...
- 2,183
0
votes
0
answers
46
views
Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
0
votes
0
answers
71
views
References on estimates for suprema of uncentered Gaussian processes?
Let $X_t, t \in T$ denote a centered Gaussian process. Let $d(t, s) = \sqrt{\mathbb{E} (X_t - X_s)^2}$.
Consider a mean function $t \mapsto \mu_t$.
Define the expected supremum
$$
S(T, \mu) = \mathbb{...
- 420
0
votes
0
answers
86
views
Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
- 671
0
votes
1
answer
113
views
Inequality on conditional variance of a vector
I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality :
\begin{equation}
p\mathbb{E}[\|X - \mathbb E [X \vert \...
- 11
0
votes
0
answers
52
views
Classifier-specific lower bounds on the misclassification rate in binary classification
Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
- 13
0
votes
0
answers
112
views
Additivity of purity of random matrix products
Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as
$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
- 2,252
0
votes
1
answer
92
views
Lower bounding the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How ...
- 405
0
votes
2
answers
118
views
the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
- 405
0
votes
0
answers
49
views
Gaussian white noise model in application
I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
- 73