# 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.

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### 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 $\|...

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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:\...

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**1**answer

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 ...

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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 $X\sim B(n,p)$.

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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

**1**answer

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 ...

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**1**answer

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} + \...

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**0**answers

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}))...

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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}(...

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**1**answer

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 ...

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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&...

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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 ...

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**1**answer

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 ...

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**1**answer

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 &...

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**2**answers

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 ...

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**1**answer

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 ...

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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 $c>1/2$. ...

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**1**answer

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 ...

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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 : \...

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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{...

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**1**answer

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?

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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 ...

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**1**answer

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}\...

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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 ...

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**2**answers

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\...

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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_{\...

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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 ...

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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 :
$$\...

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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....

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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 ...

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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_{...

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**1**answer

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 ...

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**1**answer

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 $\phi\in L^{2}(\mathbb{R},dx)$ (...

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**1**answer

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 $...

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**1**answer

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 $...

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**2**answers

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|...

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**0**answers

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 \...

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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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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 $\...

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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 ...

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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 ...

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**1**answer

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} \...

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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 ...

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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 ...

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votes

**2**answers

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 ...

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**1**answer

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**

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**1**answer

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 ...