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.
7,994
questions
1
vote
1
answer
59
views
Interpolation theorem for general rough paths
In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...
- 360
1
vote
1
answer
58
views
How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $H=(H_t, t\ge 0)$ be a stochastic process with continuous trajectories. Fix $T>0$. For $n \ge 1$, we define
$$
H_{s,n} := \sum_{i=1}...
- 375
0
votes
1
answer
50
views
Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
0
votes
1
answer
85
views
Maximal mutual information between a continuous and a discrete random variables
Let $X\sim \mathcal{N}(\mu,\sigma^2)$ be a Gaussian random variable with random mean $\mu\sim {\sf Bernoulli}(p)$, i.e., $\mu=1$ with probability $p$ and $\mu=0$ with probability $1-p$. In other words,...
- 450
2
votes
1
answer
52
views
Precise asymptotics for moments of order statistics of normal distribution
Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
- 141
0
votes
0
answers
61
views
Connectivity constant for lattices
A celebrated result due to Duminil-Copin and Smirnov states that the connectivity constant for the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$.
My question is the following: apart from the ...
- 1,393
0
votes
1
answer
43
views
Lévy measure and jump behaviour of the corresponding Lévy process
Let $(X_t)_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X_s \in A\}\rvert$
where $\Delta X_s$ ...
- 57
0
votes
0
answers
94
views
When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 253
0
votes
1
answer
83
views
A question about an intepretation of certain probability
Consider a polynomial $p(z)= \sum_0^n a_i z^i.$ In the literature there are numerous bounds about the roots of $p(z)$.Now once we prescribe certain dsitribution to the coefficients ,the bound itself ...
- 670
0
votes
0
answers
40
views
Measurability of a process defined by an integral
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{P} \subset \mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F}$ the $\sigma$-algebra generated by $\{\{0\}\times F_0\}...
- 221
1
vote
1
answer
67
views
Does the following expectation-based inequality hold?
Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...
- 63
2
votes
0
answers
31
views
Probability bounds of some ranked version of Dirichlet distribution
Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...
- 21
0
votes
1
answer
52
views
Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
- 3,092
1
vote
1
answer
53
views
Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold?
I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such ...
- 505
0
votes
0
answers
48
views
Predictability of the mild solution of a SPDE
Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
- 221
8
votes
1
answer
413
views
Popular mistakes in probability
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
-1
votes
1
answer
43
views
Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?
First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean ...
- 103
0
votes
0
answers
52
views
Vector space of random variables [closed]
Let $(\Omega, \mathfrak{S}, \mu)$ be a probability space and let $R(\Omega)$ be the space of all real-valued random variables $X: \Omega \to \mathbb{R}$ w.r.t. the above probability space having ...
- 21
1
vote
1
answer
119
views
+50
Shift-ergodic stochastic processes in continuous time
Let $\mathscr{C}:=\{\gamma : \mathbb{R}_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\...
- 187
2
votes
0
answers
40
views
Riemannian submanifolds of $2$-Wasserstein space
In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...
- 41
1
vote
0
answers
27
views
Measurability in a product space of a set defined only along its fibers
Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
0
votes
0
answers
35
views
The Rate of converging to the stationary distribution given a time in-homogenous but fast converging transition probabilities
Let $P_n$ be a sequence of transition probabilities and $X_n$ be the corresponding Markov chain. That is , $X_n=d_0P_1...P_{n}$, where $d_0$ is the initial distribution. Suppose each $P_n$ has its ...
1
vote
1
answer
58
views
Asymptotic properties of weighted random walks / infinite convolutions of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...
1
vote
0
answers
39
views
Lipschitz function of a sub Gaussian vector
I have been struggling with the following question.
Let $X \in \mathbb{R}^n$ be a $K$ sub Gaussian random vector (i.e. $\|\langle u, X \rangle\|_{\psi_2} \leq K$ for all $\|u\|=1$) and let $f : \...
- 11
0
votes
1
answer
42
views
Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?
Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?
-1
votes
0
answers
57
views
What is the probability that after throwing n balls into k bins uniformly we will have a different number of balls in each bin?
I want to know if it is possible to compute the following problem (Or at least give an estimation on the lower bound) :
Given $n$ balls and $k$ bins where $n>>k$, we throw $n$ balls into those $...
0
votes
0
answers
24
views
Fluctuation-dissipation theorem for Markov processes
In the context of particle systems of non-gradient types (see e.g. here, Step 2 on page 633), I recently encountered the concept of fluctuation-dissipation theorem (FDT). Since it is a major result in ...
- 83
0
votes
0
answers
37
views
Expected value larger than conditional expected value
Let $X$ and $Y$ be two random variables, with $\mathbb{P}[X] > 5 \cdot \mathbb{P}[Y]$.
Let $h_X(\Delta)$ and $h_Y(\Delta)$ donate the number of hits in a histogram after a time period $\Delta$. In ...
3
votes
1
answer
88
views
First time random sum exceeds value
Suppose $X_n$ $n = 1, 2, \ldots$ are i.i.d random variables with $\mu := \mathbb{E}[X_n]$ > 0. (although they are not necessarily non-negative). Then if $S_n = \sum_{k=1}^n X_k$ and $\tau_a$ = $\...
- 33
0
votes
1
answer
66
views
Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
- 109
3
votes
1
answer
79
views
Probabilistic method Alon and Spencer Azuma's inequality
Theorem 7.5.2 states:
Let $v_1, \dots, v_n$ be vectors with $\|v_i\| \leq 1.$ Let $\epsilon_1, \dots, \epsilon_n \in \{-1, 1\}$ be independent with uniform probability and let $X=\|\epsilon_1 v_1 + \...
0
votes
0
answers
70
views
Control the largest eigenvalue of random matrix
Objective: If there exists $\epsilon > 0$ such that $\sigma <
\sqrt{\frac{n}{(2+\epsilon)\log n}}$
then, with high probability, we have $\lambda_{\text{max}}(D_{[-W]}+W)<\frac{n}{\sigma}$. $W$...
- 43
1
vote
0
answers
20
views
Convergence of eigenvalues in the bulk of the GbetaE?
Let $\lambda_{k(n)}$, $k(n) / n \to a \in (0,1)$, be an eigenvalue in the bulk
of the $n \times n$ Gaussian Unitary Ensemble (GUE) normalized so that its spectral measure converges to the semi-circle ...
- 91
1
vote
1
answer
61
views
An inequality relating $\ell_1$ distance of input and output of a Markov krnel
Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$.
Let $\mu$ and $\nu$ be two probability measures ...
- 1,079
2
votes
1
answer
212
views
Lower bound on sum of independent heavy-tailed random variables
I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...
- 23
3
votes
1
answer
88
views
Inequality: multivariate normal distribition
Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$
Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>...
- 221
4
votes
1
answer
255
views
Joint distribution of minor of Wigner Hermitian matrices
Let $A$ be an $n\times n$ random matrix with i.i.d entries (say standard Gaussian) $A_{ij}$. I know that there is a CLT type result known for the determinant of $A$. More precisely there is a CLT for $...
- 351
0
votes
0
answers
17
views
Bound on inverse covariance from covariance in regularized covariance estimation problem
In this paper by Bickel and Levina, I am confused about result (A15) which claims that since
$$
(A14) \qquad \| \text{Var}(\mathbf{X}) - \widehat{\text{Var}}(\mathbf{X})\|_{\max} = O_P(n^{-1/2} \log^{...
- 176
0
votes
1
answer
87
views
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...
- 375
3
votes
1
answer
109
views
Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...
- 133
-1
votes
0
answers
33
views
Convergence of a non-iid series
Let $\mathbf y \in \mathbb C^N$ be a gaussian random vector with covariance matrix $C_{\mathbf y}$, then $\mathbf x = C_{\mathbf y} ^\frac{-1}{2} \mathbf y$ will be an i.i.d random vector. For a given ...
- 13
7
votes
1
answer
175
views
Counting returns in null-recurrent random walk
Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N_1(t)$ (resp. $N_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker ...
- 17.9k
2
votes
2
answers
86
views
Density of $W_t$ assuming it stayed above a line $L$
Let $W_t$ be a Wiener process with $W_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).
Assume that $W_t$ stayed above $L$ up to time $T$. What is ...
- 2,718
1
vote
0
answers
107
views
Strong law of large numbers when the largest value is deleted
If $X_1, X_2, \ldots X_n, \ldots$ are iid integrable real random variables, and $S_n = X_1 + \cdots + X_n$,
$n \geq 1$, the standard strong law of large numbers expresses that $\frac 1n S_n \to E(X_1)$...
- 11
1
vote
1
answer
97
views
Stochastic integral with non-anticipating integrand
Let $B$ be a Brownian motion. We want to define $$ \int_{0}^{t} B_{s} dB_{s} : = \lim_{n \to \infty } \sum_{k = 1}^{2^{n}t} B_{\frac{k-1}{2^{n}}}[ B_{\frac{k}{2^{n}}} - B_{\frac{k-1}{2^{n}}}]. $$
To ...
- 31
6
votes
1
answer
99
views
What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map?
A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. ...
- 75
2
votes
0
answers
50
views
Finding the limit of a specific sequence of linear processes
Given a strictly stationary random process $(\xi_t)_{t \in \mathbb Z}$. Define $\mu:= E[\xi_t]$, for all $t$. Suppose $(\xi_t)_{t \in \mathbb Z}$ ergodic:
\begin{equation}\label{a}\tag{E}
\frac{1}{n}\...
- 155
8
votes
1
answer
446
views
Scheduling "parent talks" at school
Real life motivation. In my younger son's class, there are $18$ students. His teacher provided $18$ time slots for the parents of each child to have a 30-minute conversation of their kid's progress in ...
- 41.4k
0
votes
0
answers
40
views
Objective function in denoing diffusion probability model
The objective of the diffusion model can be written as follow (from Lil'Log):
\begin{aligned}
L_\text{VLB}
&= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}...
- 1
4
votes
1
answer
216
views
Random walk visiting a cylinder infinitely often
I wonder whether a $d$-dimensional random walk $S_n$, generated by the infinite i.i.d. copies of X given by:
$X=e_1=(1, 0, 0, ..., 0)$ (with probability $p_1$)
$X=e_2=(0, 1, 0, ..., 0)$ (with ...
- 91