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.
0
votes
0answers
4 views
Concentration of the norm of a random vector
Suppose $X$ is a random vector denoted as $(X_1,\cdots,X_n)$, where $X_1,\cdots,X_n$ are iid random variables with sub-Gaussian distributions. For all $i$, suppose $E[X_i^2]=1$ for simplicity and $\|...
0
votes
0answers
22 views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
0
votes
1answer
58 views
Strictly Proper Scoring Rules and f-Divergences
Let $S$ be a scoring rule for probability functions. Define
$EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$.
Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...
4
votes
3answers
75 views
A clean upper bound for the expectation of a function of a binomial random variable
I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
-4
votes
0answers
26 views
Probability and profit [on hold]
I have the following problem:
I am taking trades.
The probability that I win any given trade is 0.6.
The probability that I lose any given trade is 0.4.
If I win 3 times in a row, I win 7.
If I win ...
1
vote
1answer
50 views
Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors
Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
2
votes
1answer
51 views
Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality
Let $X$ be a random variable s.t. for $v, b > 0$ and $C \geq 1$:
$$ P(X \geq t) \leq C\exp\left(-\frac{t^2}{2(v^2 + bt)} \right) $$
I am trying to show that $\mathbb{E}X \leq 2v(\sqrt{\pi} + \...
-1
votes
0answers
46 views
Numerical expectation involving Dirac-delta function
I'm looking for the way for numerical integration including Dirac-delta function. Here is what I want to obtain in numerical way such as Monte Carlo sampling.
$$ \int m(\mathbf{x})\delta(G(\mathbf{x}))...
0
votes
0answers
35 views
Computational Time Complexity bounds for approximate maximum of a sequence/array
The problem I have is the following: Given a sequence $x_1, \ldots, x_N$ for $N$ very large. For any $\varepsilon, \delta > 0$, find a number $\hat{x}_{\varepsilon, \delta}$ such that
$$\mathbb{P}(...
0
votes
1answer
78 views
How to find a special random variable? [on hold]
Suppose random variables $X_1$ and $X_2$ have the same distribution under P, $Y_1$ is an arbitrary random variable,let $Z_1:=X_1+Y_1$.Can we find a r.v. $Y_2$ which has same distribution as $Y_1$,such ...
-2
votes
0answers
58 views
Joint PDF of Laplace distribution and Gaussian Distribution
If $X$ follows a Laplace distribution with PDF
$f(x\left| {\mu ,b} \right.) = \frac{1}{{2b}}\exp \left( { - \frac{{\left| {x - \mu } \right|}}{b}} \right)$
where $\mu$ is a location parameter and $b&...
2
votes
0answers
56 views
Skorokhod representation for weak convergence of exchangeable arrays
Let $(X_e)=(X_e)_{e\in \mathbb{N}^{(k)}}$ be a $k$-dimensional exchangeable real random array (see this note for the definition), where $k\in \{1,2,\ldots \}$ is fixed and $\mathbb{N}^{(k)}$ denotes ...
1
vote
1answer
61 views
Expected size of matchings in a cubic graph
Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...
3
votes
1answer
88 views
Mixing time and spectral gap for a special stochastic matrix
Conisder the following dimension stochastic matrix,
\begin{bmatrix}
p & q & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 &...
10
votes
2answers
488 views
Theorems like the Lovász Local Lemma?
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have ...
3
votes
1answer
80 views
concentration inequality for a weighted sum of independent but not identical binary variables
Let $\alpha\in[0,1]$ be a fixed constant, and let
$w,x\in[0,1]^n$ be two vectors such that $\sum_i w_i x_i=\alpha$.
Define $Y = \sum_i w_i X_i$, where $X_i \sim \operatorname{Bernoulli}(x_i)$, so it ...
1
vote
0answers
40 views
Monotonicity of $\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c \right]$ in $n$ for $c>\mathbb{E}(X).$
The nice question below was answered in the affirmative
in On the sum of uniform independent random variables
Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. ...
2
votes
1answer
62 views
Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture
Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
1
vote
0answers
47 views
About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
1
vote
2answers
137 views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
2
votes
1answer
177 views
Statement about independence of random variables
If I have 2 random variables $\xi, \eta$ and $\forall n,m \ \mathbb E\xi^n\eta^m=\mathbb E\xi^n \mathbb E\eta^m$, does this imply that $\xi,\eta$ are independent? How to show it?
0
votes
0answers
21 views
Reference: Stochastic Filtering Infinite Dimensions
I've come across these Hilbert Space Signal Finite Dimensional Measurements and Linear Gaussian Hilbert space signal and measurements.
Is there any literature solving the Zakai equation when both ...
3
votes
1answer
65 views
Gaussian expectation of outer product divided by norm (check)
I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix:
$$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...
-2
votes
0answers
25 views
Looking for a generalization of Binomial distribution and it's properties
In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A ...
3
votes
2answers
76 views
Expected minimum of a linear function on the unit cube
Let $c\in\mathbb{R}^n$ and let $X_1,X_2$ be two independent uniform samples on the unit cube in $\mathbb{R}^n$. Is there anything at all (in an analytic sense) we can say about the expectation $E\min\...
2
votes
0answers
44 views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
1
vote
0answers
82 views
Clarification about the ϵ -net argument
I have been reading the paper Do GANs learn the distribution? Some theory and empirics.
In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...
0
votes
0answers
32 views
Asymptotic upper densities in infinite binary stochastic processes
Consider an infinite binary process $X=X_1,X_2,\ldots$ (with corresponding probability $P$). For some bits $1$ is less probable than $0$. I am interested in the following asymptotic upper density :
$$\...
21
votes
0answers
393 views
How many random walk steps until the path self-intersects?
Take a random walk in the plane from the origin,
each step of unit length in a uniformly random direction.
Q. How many steps on average until the path self-intersects?
My simulations suggest ~$8....
3
votes
0answers
59 views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
-1
votes
0answers
43 views
Identical push-forward but not stationary
I'm having some trouble coming up with a counter-example for this problem:
Give an example of a stochastic process $\{X_n : n \in \mathbb{Z}^+\}$ on $(\Omega, \mathcal{F}, P)$ such that $P_{X_n} = P_{...
1
vote
1answer
52 views
Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix
Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent)
what is the distribution of ${y^T M y}$?
is there a high probability bound on $|{y^T M y}|$?
Most ...
0
votes
1answer
121 views
Divergence form degenerate pde and Feynman Kac
Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
1
vote
1answer
54 views
Convergence of probability density function
There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...
4
votes
1answer
50 views
Rate of decay in the multivariate Central Limit Theorem
The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S_n=X_1+\cdots+X_n$ of independent random variables $...
6
votes
2answers
146 views
A limit obtained from a probability distribution on the positive integers
Let $p_n$ be a probability distribution on the positive
integers $n$. Let
$$ \frac{1}{1-\sum_{n\geq 1} p_nx^n}=\sum_{k\geq 0}a_kx^k. $$
Suppose there does not exist an integer $d>1$ such that
$d|...
3
votes
0answers
126 views
Central Limit Theorem for simultaneous sums
Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \...
5
votes
1answer
74 views
Variance of sum of $m$ dependent random variables
I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.
Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
1
vote
0answers
38 views
Gilbert-Varshamov with weight condition
For $N$ integer, it is known (Gilbert-Varshamov) that there exists a subset $S$ of $\{0,1\}^N$ such that $|S|$ is still exponential in $N$ and such that two elements $x, x'$ in $S$ are at least ...
1
vote
0answers
92 views
Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?
Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...
0
votes
0answers
52 views
Probability of detecting small bias in a die in the low confidence regime
We are given a biased $m$-sided die: one of the sides has probability $\frac{1}{m} + \gamma$ and all the rest have probability $\frac{1}{m} - \frac{\gamma}{m-1}$ each. The goal is to figure out which ...
2
votes
1answer
71 views
How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set?
I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\...
3
votes
0answers
165 views
How should the proof of the XYZ theorem be understood?
The XYZ Theorem of Shepp [1] states that for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for any three ...
0
votes
0answers
39 views
Mean first-passage time for a marked Poisson process
Given a marked Poisson process in one dimension
$$
Y(t)=\sum_{\{t_i,a_i\}}g(t-t_i,a_i)
$$
so that $Y(t)$ is a sum of impulses arriving as a Poisson process and the impulses $g$ belong to a family of ...
8
votes
1answer
106 views
Log-concavity of repeated convolution
Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
0
votes
0answers
49 views
Generalized eigenvectors of block triangular random matrices
Let $A = (a_{ij})_{1 \leq i,j \leq N}$ and $B = (b_{ij})_{1 \leq i,j \leq N}$ be random matrices, with each $a_{ij}$ and $b_{ij}$ an independent random variable with continuous density function, zero ...
-1
votes
0answers
31 views
One sided point-wise Berry-Esseen like inequality for discrete variable
Let us consider a distribution $\mathcal{L}$ on a finite set of integers
(actually I'm even happy with any RV on the set $\{-1,0,1\}$) (with probability not to extremes : roughly around 1/4 for -1, 1 ...
2
votes
2answers
149 views
Birthday problem extension to unequal probabilities and multiple collisions
Let $p_1, ... ,p_k$ denote the probabilities of drawing bin $1, .. ,k$, where $\sum_{i = 1}^{k} p_i= 1$. My question is if we draw $n$ times, how can I show that the probability that all bins are ...
5
votes
1answer
141 views
Binomial Distributions and Inequality
Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...
-1
votes
1answer
99 views
Distribution of first time a 1D random walk hits n or -n
Let $(\omega_1, \omega_2, \ldots)$ be iid in $\{-1, 1\}$ and $X_k = \sum_{i=1}^k \omega_i$ be a simple one-dimensional random walk.
Let $\tau_n = \min \{i\in\mathbb{N}: |X_i|=n\}$ be the first time ...
