All Questions
9,498 questions
0
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62
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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 ...
- 211
0
votes
0
answers
268
views
Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
- 250
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0
answers
140
views
Reference for convergence to a Poisson Point Process
Edited after comment by Ofer Zeitouni
I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$,
\begin{equation}S_n(i)=\sum_{j=...
- 66
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votes
0
answers
89
views
Why there isn't lexicographically smallest solution to a bounded linear program?
I am learning computational geometry when I run into this confusion. "A bounded 2D linear program may not have a lexicographically smallest solution", as the book says. I wonder why? I think we can ...
- 1
0
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0
answers
424
views
Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
- 31
0
votes
0
answers
103
views
Expectation of maximal Wasserstein distance between empirical distribution and a pdf
Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $
\hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$.
Do ...
- 109
0
votes
0
answers
57
views
Absolute continuity of probability measures determined by dependence structure
We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
- 1,095
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votes
0
answers
77
views
Ladder times of a Brownian motion with drift
Let $(B_s)_{s_\geq 0}$ be a standard Brownian motion and fix $t>0$. For $u>0$, set $T_u=\inf\{s>0, B_s+s t>u\}$. Now consider $x>0$ such that $\sup_{0 \leq s \leq x} (B_s+st)=B_x+xt$ ...
- 406
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votes
0
answers
268
views
Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile
$F(x)$ and $G(y)$ are distribution functions.
Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as
$$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$
and
$$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
- 141
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votes
0
answers
139
views
Statistical models of functions
I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...
- 1,313
0
votes
0
answers
84
views
If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?
Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...
- 167
0
votes
0
answers
691
views
$L_1$ convergence for a product of indicator functions
Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions
$$
\lim_{N\...
- 479
0
votes
1
answer
270
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 ...
- 173
0
votes
2
answers
308
views
Predictability of countably valued accessible stopping times on complete and cadlag filtrations
The following question is motivated by this part of the proof of Lemma 2 on page 107 of the book Stochastic integration and differential equations of Philip Protter.
Lemma 2. Let $T$ be a totally ...
- 69
0
votes
0
answers
219
views
Question about Protter's proof of the Ito's formula
The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential ...
- 69
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
- 205
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votes
0
answers
29
views
Probabilistic timed automata transition
I am kind of new to timed automata and I have a question related to their correctness and synchronisation.
Assume that I have three states, A, B and C. I have also two clocks, $x$ and $y$ that are ...
- 39
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votes
0
answers
119
views
the enumeration of 2 dimensional lattice walks with fixed number steps and largest distance from the end point ti the origin
There is actually an one dimensional version of this problem. For each step of the lattice walk, we can move either east for one unit or west for one unit. The problem is that given a fixed $n$ steps ...
- 167
0
votes
1
answer
565
views
Continuity w.r.t time vs Continuity w.r.t. stopping times
Several places in "Optimal Stopping and Free-Boundary Problems" Peskir and Shiryaev make the assumption that a (Markov) process $X = (X_t)_{t\geq 0}$ has sample
paths which are right continuous and ...
- 13
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0
answers
165
views
Probability that the perturbed convex hull is larger than the original one
I am wondering if any convex geometers/probabilists have looked at the following question:
Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
- 133
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votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
- 13.2k
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votes
0
answers
72
views
Invariant measures for a renewal process driven by Interarrival times bounded away from zero
Good morning, I apologize in advance if my question sounds too basic but after some research I was unable to come up with satisfactory answers to my doubts.
I am currently studying a model which ...
- 277
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votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
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votes
0
answers
52
views
Birth and death process, inequality between two stopping times
I have a discrete times birth and death process $\{\Psi_n\}_{n\in \mathbb N}$ with birth probability $p$ and death probability $q$ defined as follows:
\begin{align}
\Psi_n=\sum_{i=1}^n\eta_i
\end{...
0
votes
0
answers
76
views
Ornstein-Uhlenbeck type process with thresholding
(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...
- 31
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votes
0
answers
69
views
Convergence of a stochastic process in probability
I came across the following. For any fixed $n$, let $\{X_{n}(s) \}_{s\geq0}$ be a stochastic process and let $\{B_n(s) \}_{s\geq0}$ be a Brownian motion. We wish to study the behaviour of $\{X_{n}(s) \...
- 101
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votes
0
answers
167
views
Prime gap heuristics (follows up my question "Moments of merit")
I previously asked generally what people knew or conjectured concerning the moments of the probability distribution governing $M_n:= g_n/\ln(p_n)$, the normalized $n$th prime gap (or ``merit''). Greg ...
- 17.6k
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votes
0
answers
72
views
Generating function for number of r-disjoint subsets each of size k
Fix $n, k$. Let
$$
C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!}
$$
be the number of ways to form $r$ disjoint subsets each of ...
- 101
0
votes
0
answers
64
views
Probability of collision of sums of vectors
Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function.
Does there exist a random matrix $R \...
- 321
0
votes
0
answers
368
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
0
votes
0
answers
112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 505
0
votes
0
answers
129
views
Convergence of sequence of stopped partial sums to functional of Wiener process
I asked the same question on stackexchange (with less details) and decided to post it here. Hopefully it's close to the research level.
Preliminary: I have a sequence of normalized partial sum ...
- 223
0
votes
1
answer
279
views
Expected properties for a PDE whose solution is supposed to be something that doesn't exist
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
- 247
0
votes
0
answers
35
views
Iterating the Voter protocol
Assume you have an array of length $n$ filled with the numbers $1,2,...,n$. (Actually, it only matters that all numbers are different.) This corresponds to a Dirac delta distribution for the number ...
- 4,829
0
votes
0
answers
130
views
Average of Tracy-Widom distribution
I have posted this to MSE, but it got no attention (https://math.stackexchange.com/questions/2619324/average-of-tracy-widom-distributions)
The Tracy-Widom distributions famously describe the ...
- 1,549
0
votes
0
answers
80
views
Not exactly directed percolation
Is the following problem known/well-studies? I'm looking for references or a name that I can look up.
I start with $N$ cell, each one divides into two cells, each one of the new cells either dies ...
- 371
0
votes
0
answers
90
views
criterions for polar set of Feller processes
Suppose $X_t$ is the solution to
$$
d X_t=b(X_t)dt+dL_t,\quad X_0=x.
$$
where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.
Assume $\Gamma\subseteq ...
- 515
0
votes
0
answers
57
views
Fixed Length Gaussian pdf
In a research problem about detection theory I faced with the following question.
How can I find the conditional Gaussian probability density function
$$ f({\mathbf w} \; | \; \|{\mathbf w}\|^2)$$
...
- 273
0
votes
0
answers
92
views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
- 141
0
votes
0
answers
102
views
Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$
Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.
I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.
Specifically, if $\delta,p=o(1)$ are not ...
- 618
0
votes
0
answers
141
views
Probability of getting through with a phone-call
Alice is quite popular. She gets called on her cell-phone in a Poisson$(\lambda)$ manner. She answers her calls when possible, and ignores them when in the middle of conversation. Since you know her ...
- 396
0
votes
0
answers
250
views
Can we make two random variables independent at a low cost?
Let $X$ and $Y$ be two discrete random variables with joint probability mass function $p(x,y)$ such that
$$\|p(x,y)-p(x)p(y)\|_1=\sum_{x\in\mathcal{X},y\in\mathcal{Y}}|p(x,y)-p(x)p(y)|\leq\epsilon$$
...
- 287
0
votes
0
answers
137
views
Expected Number of Triangles
A unit square is divided up with $n$ random lines. The random lines are chosen as follows, we choose one side of the square and pick a random point on that side. From there we choose a random point on ...
0
votes
0
answers
93
views
Changing Couplings of Discrete Random Variables
Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
- 329
0
votes
0
answers
169
views
Behaviour of a Markov Chain, given a Lyapunov condition
I'm reading this notes from Martin Hairer about convergence of Markov Processes (on a discrete state space $S$ and in continuous time). On page 12, before presenting the so-called "Harris Theorem", ...
- 203
0
votes
0
answers
75
views
Drunkards Uphill Walk revisited
We want to help the poor git...
Old Question
...with a bias to speed up things. We replace step 1
1. Draw ball, memorize color, throw it back.
with
1a-c. Draw ball, memorize color, throw it back. ...
- 4,829
0
votes
0
answers
72
views
A random variable standing for the size of connected component including a given node in a tree
Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for ...
- 1,551
0
votes
0
answers
65
views
Wanted: example of a non-stationary sequence with reverse empirical measure
Assume we have a sequence $\xi=(\xi_1,\xi_2,\dots)$ of random variables such that $$\eta=\left(\frac{\sum_{i=1}^n \delta_{\xi_i}}{n}\right)_{n\geq 1}$$ is a reverse-martingale with respect to its own ...
- 211
0
votes
0
answers
141
views
Effect of partitioning the realizations of random variables on the total variation distance?
Let $X$ and $Y$ be two random variables with joint pmf $p(x,y)=p(x)\cdot p(y|x)$ and $X$ has uniform distribution. Also assume that the following relation is satisfied:
\begin{align}
\lVert p(y|x)-p(y)...
- 287
0
votes
1
answer
80
views
Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v
Given
\begin{equation}\label{eq:definition_of_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\
{z}_{21} & {z}_{22} & \cdots & {...
