Tagged Questions
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
2
votes
1answer
57 views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
0
votes
0answers
38 views
$L_1$ convergence for a product of indicator functions
Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions
$$
\lim_{N\...
0
votes
0answers
52 views
Joint distribution of a drifted Brownian motion and its supremum
let $X_t= W_t + bt$ be a drifted Brownian motion on $(\Omega, \mathcal{F}, \mathbb{P})$. Given the stopping time process $T_a=\inf\{t \geq 0 : X_t=a\}$ with $a>0$, how can I compute $\mathbb{P}\{...
-1
votes
0answers
25 views
Application of non-commutative Khinchine inequality
I am looking for applications of non-commutative Khinchine inequality (see below) in case when Rademacher random variables are tight by the condition $\sum_{i=1}^N\varepsilon_i=M, \, -N \leq M\leq N$....
3
votes
0answers
19 views
product of right continuous filtrations is right-continuous?
Let $\mathcal{G} = \sigma \lbrace G_{1},..., G_{n} \rbrace$ where $G_{1},..., G_{n}$ are subsets of $\Omega_{1}$ and $(\mathcal{F}_{t})$ is a right-continuous filtration on $\Omega_{2}$. Is $(\mathcal{...
0
votes
0answers
22 views
(Upper) bounding the MGF of a semi-decoupled non-homogeneous Rademacher chaos of order 4
Let $(\xi_i)_{1\leq i\leq n}$, $(\xi'_i)_{1\leq i\leq n}$ be two independent vectors of independent Rademacher random variables, and
$(a_{ij})_{1\leq i,j\leq n}$, $(b_{ijk})_{1\leq i,j,k\leq n}$, $(c_{...
3
votes
1answer
103 views
non-backtracking random walk in regular (finite) graphs
I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a ...
1
vote
1answer
82 views
Expected value of sin(X) for Gamma r.v. X in closed form (approximation is fine)
I have a random variable $X \sim \operatorname{Gamma}(\alpha, \beta)$.
How can I compute or approximate $\mathbb{E} \sin(X)$ very quickly? Iterative quadrature would be too slow, I need some closed ...
4
votes
1answer
70 views
+50
Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
0
votes
0answers
78 views
Is there a probabilistic proof/interpretation of Mergelyan Theorem
I came across Mergelyan's Theorem:- Let K be a compact subset of the complex plane C such that C∖K is connected. Then, every continuous function $f : K \to C$, such that the restriction f to int(K) ...
2
votes
0answers
127 views
Sum of Gaussian matched by Brownian Motion?
Given independent Gaussian $d$ dimensional vectors $G_i$,
If $\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T=n \cdot I_{d \times d} + o(n^{1-\epsilon})$.
there exists Brownian motion $...
-1
votes
0answers
36 views
Ergodic sequence joint probability rate to zero
Let $X_1,X_2,\ldots$ be a stationary ergodic sequence of random variables and let $A\in\mathbb{R}$ be such that $0 < P(X_1\in A)< 1$. Then
$$
\lim_{m\rightarrow\infty}P\left(\bigcap_{n=1}^{m}X_n\...
1
vote
0answers
34 views
Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball
Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [...
6
votes
4answers
226 views
Improvement of Chernoff bound in Binomial case
We know from Chernoff bound
$P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where
$X$ follows Binomial($N, \frac{1}{2}$).
If I take $N=1000, \epsilon=0.01$, the upper bound is ...
13
votes
3answers
706 views
Probability of commutation in a compact group
It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...
-2
votes
0answers
55 views
probability of grid back to starting point after 4 steps [closed]
Starting from point (0,0), there are 5 choices for each step,
staying at the same coordinate
go left (x-1)
go right (x+1)
go up (y+1)
go down (y-1)
What is the probability of returning to starting ...
0
votes
1answer
33 views
Right tail decay of F distribution [closed]
Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?
-1
votes
1answer
67 views
Expected value of $W_{t_i} W^2_{t_{i+1}}$
I stuck in determining the expected value of the following product
$E[W_{t_i}W_{t_{i+1}}^2]$ where $W_{t_i}$ and $W_{t_{i+1}}$ are Brownian with normal distribution, i.e. $W_{t_i}\sim N(0,t_i)$. I ...
3
votes
3answers
153 views
What is the probability distribution of the $k$th largest coordinate chosen over a simplex?
Suppose we're selecting points uniformly at random from the $N$-simplex
$S_N = \{x \in \mathbb R^{N+1}: $ all $ x_i \ge 0$ and $x_1 + \ldots x_N = 1\}$.
One way to do this in practice is choose $N-...
-3
votes
0answers
25 views
probability of success to reach to d distance in N steps while following random walk with different step size [closed]
How can we find the probability of success to reach to destination at 'd' distance from right while following random walk with large step size in right direction and small step size in left direction, ...
1
vote
2answers
100 views
lower bound the probability of at least L collisions
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...
4
votes
1answer
86 views
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 \...
2
votes
1answer
63 views
Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
4
votes
0answers
202 views
What is the category of algebras for the finitely supported measures monad?
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...
3
votes
0answers
90 views
Galton Watson tree with various kinds of offspring
As far as I understood, for the Galton-Watson tree process, the offspring are of one type. I am thinking of the case where we have offspring of different types. I have illustrated this in the example ...
-1
votes
0answers
67 views
An application of the contraction principle
Let $ 1<a<b$ and denote by $P_{p}(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$ with finite p-th order moment, endowed with the Wasserstein topology.
If $(\mu_{n})$ ...
-1
votes
0answers
22 views
Characteristic function of hitting time
Suppose that $X_t$ is an affine process, $f$ is a convex function with values in $\mathbb{R}$ such that $X_0=0$, and $M>0$. Then what can be said about the hitting time
$$
\tau \triangleq \inf\...
11
votes
3answers
462 views
A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I ...
3
votes
1answer
103 views
Log concavity of the maximum of dependent Gaussians
Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
2
votes
0answers
39 views
Convergence to the probability generating function of a Poisson process
I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that
$\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...
4
votes
1answer
297 views
What is the six positive real number for a dice producing a highest chance?
Say there is a dice with six faces, each face has a positive real number different from others. There is a chessman on the origin of the number axis. In each trial, the dice rolls infinite times. ...
2
votes
1answer
85 views
Convergence of a stochastic sequence?
I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...
0
votes
1answer
76 views
Is the normal product distribution sub-gaussian?
Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
1
vote
0answers
47 views
A Random Graph Process
I'd like to understand the following random graph process. I'm not sure if it's difficult or straightforward, so apologies if this is below the level of mathoverflow, but I've gotten no response on ...
3
votes
1answer
141 views
Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?
I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...
1
vote
0answers
66 views
Schilder's theorem for brownian bridges
I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the ...
3
votes
1answer
110 views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
3
votes
1answer
130 views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
4
votes
0answers
118 views
Characterization of KL divergence for continuous variables?
This is an analog of an older question:
What characterizations of relative information are known?
With the modification that I’m interested in the case when the distribution is over something that’s ...
1
vote
0answers
51 views
Negative association in a “k out of n” process
Suppose we have $n$ distinct balls labeled $1,2,\dots,n$ in a black box. Now we want to fetch $k$ balls from the box, one by one. Let event $E_i$ be that the label of the $i$-th ball we fetch is no ...
6
votes
0answers
124 views
What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
2
votes
0answers
55 views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
0
votes
1answer
52 views
Bivariate Poisson-Binomial distribution
Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
4
votes
0answers
98 views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
2
votes
1answer
37 views
Convergence of $\sup_{l \leq \rho \leq u} | \frac{1}{\sqrt{n}} \sum_{i=[\rho n]}^n \sum_{k=1}^i i^{-1} x_i y_k |$ in probability to zero
I am dealing with a sequence $\{(x_i,y_i)\}$ of zero-mean random variables. For simplicity we can assume that the sequence is i.i.d. Define $Y_i := i^{-1} \sum_{k=1}^i y_k$. I would like show that
\...
6
votes
3answers
314 views
Connections between martingales and Fourier analysis
I have had this strange feeling recently that somehow, the theory of martingales we study in probability, and the theory of Fourier analysis are very alike. But I am not able to formalize my thoughts.
...
1
vote
0answers
40 views
Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support
Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
7
votes
1answer
291 views
The probability that two elements of a finite nonabelian simple group commute
It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
2
votes
0answers
59 views
Covering a sphere with ellipsoids in high dimension
For $k\times k \ \Sigma\geq 0$, $n$ large, fix $E:= \{(x_1,\dots, x_n): \frac{1}{n}\sum_i x_i^\dagger \Sigma x_i \leq 1\}$. Fix $(z_m)_m$ as $M$ points iid uniform on $\mathbb{S}^{nk-1}\subset \...
2
votes
1answer
79 views
Tail condition (Varadhan's lemma)
I would like your help with the following tail condition, which arises in the theory of large deviations.
Let $P(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$, $ G:P(\mathbb{...
