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.
8,631
questions
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A queuing process where customers must be detected
Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of "...
4
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1
answer
2k
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Expectation of the time t standard brownian motion stopped at itself's square
I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...
4
votes
3
answers
906
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simultaneous jumps of independent Levy processes
Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.
Is anyone familiar and can refer me to a result(or a counterexample) which states that
${\displaystyle \sum_{0\leq ...
4
votes
1
answer
154
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diffusions corresponding to estimators
I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
4
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1
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2k
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Multivariate Central Limit Theorem For Non-Identical Distribution [closed]
Among the different generalizations of the CLT available on the web, I found these
CLT for the sum of non-identical (and independent) random variables
CLT for the sum of identical (and independent) ...
4
votes
2
answers
1k
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Probability distribution over cluster size in Erdős–Rényi random graph.
My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph (...
4
votes
1
answer
2k
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Probability $k$ bins are non-empty.
The following problem arises in the analysis of Bloom filters.
Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at random and asks if ...
4
votes
1
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165
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Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
I have two bags: one filled with red marbles, and one filled with blue marbles. I would like to fill a bin with only $k$ red marbles and no blue marbles. However, I can only sample (with replacement)...
4
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2
answers
306
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Sampling from a recursively defined distribution
I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...
4
votes
1
answer
878
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distribution of the number of empty bins in a multinomial setting
Let $(X_1,X_2,\ldots,X_k)$ be a multinomial random vector with parameters $n, p_1, p_2, \ldots, p_k$ (i.e., we throw randomly $n$ balls into $k$ bins, so that for each ball, the probability of landing ...
4
votes
1
answer
416
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"birds on wire" type problem
Consider $n$ individuals {$1,2, \ldots, n$}. For each (unordered) pair of individuals $i \neq j$ we consider a random variable $X_{i,j}$ that can be thought of as the distance between $i$ and $j$. ...
4
votes
2
answers
3k
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Inversion of Moment-generating functions (aka Laplace transform of prob dist)
I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by
\begin{equation}
M_X(t) = \text{E} \exp(tX)
\end{equation}
Since I have ...
4
votes
1
answer
777
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Maximum entropy probability distribution with known quantile
For continuous distributions on x>0 with known mean m, the exponential distribution f(x) = (1/m)exp(-x/m) is the maximum entropy distribution, with entropy H(f) = ln(m)+1. I have a problem where I ...
4
votes
2
answers
414
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Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
4
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3
answers
2k
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Probability of overlapping of repetitive events
The question is to compute or estimate the following probabilty.
Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If ...
4
votes
2
answers
868
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Puzzle in Martin Gardner book [closed]
What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? ...
4
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2
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1k
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BM and interpretation of stopping time sigma algebra
Suppose $H$ and $K$ are open subsets of $\mathbb{R}^d$ containing the origin with $H\subset K$, $B_t$ a standard Brownian motion starting at the origin, $\mathcal{F}_t$ its canonical filtration, and $\...
4
votes
2
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652
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# bridges in random connected graph
Suppose we have an Erdos random graph with $n$ vertices and $c n$ edges.
What can you say about the probability that the graph is connected?
(More importantly) If it is connected, what is the ...
4
votes
1
answer
340
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approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
4
votes
1
answer
379
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initial condition of a diffusion approximation
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
4
votes
1
answer
154
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Probability problem in Sheehan's conjecture
As my first math project, I have been working on Sheehan's Conjecture
and am stuck for weeks. I wonder if I am at a dead end.
Sheehan's Conjecture states that every Hamiltonian 4-regular simple
graph ...
4
votes
1
answer
342
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Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution
Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$.
Question. What are necessary and sufficient conditions on $Q$ to ensure ...
4
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1
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212
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Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...
4
votes
1
answer
309
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Minimum of random walks
Let $M$ independent and identical random walks that follow the chi-squared distribution, i.e. in each step, a $X^2_1$ random variable is added. I am interested in the minimum random walk at each step. ...
4
votes
1
answer
377
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A process of repeated convolution and conditioning and the resulting sequence of probability distributions
I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$.
Let $D_1$ be the $n$-dimensional Gaussian distribution with ...
4
votes
1
answer
288
views
Large deviations for sequences that are not sums of iid
Suppose $(S_n)_n$ is a sequence of real random variables. Denote their cumulant generating functions by $K_n(t) = \log\mathbb{E}\left[\mathrm{e}^{t S_n}\right]$, and assume that each $K_n$ is finite ...
4
votes
1
answer
203
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Schauder basis of the Hardy space of semi-martingales
Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm
$$
\...
4
votes
1
answer
80
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techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$
We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...
4
votes
1
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254
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How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?
I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging.
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...
4
votes
3
answers
302
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Probability that $k$ random subsets of a fixed size covers a set
Let $A=\{1,\ldots,n\}$. Now, we uniformly randomly select $k$ subsets, $A_i$ of size $d$ from $A$. What is the probability that $\bigcup_i A_i=A$? This seems to be natural variant of the set cover ...
4
votes
2
answers
235
views
Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
4
votes
2
answers
308
views
Concentration of $k$-th pairwise distance of random points in a unit square
For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
4
votes
1
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472
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Intuition behind the noncentral chi square as Poisson mixing
It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).
...
4
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3
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359
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Does the Green's function of the simple random walk on $\mathbb Z^d$ always vary locally?
Let $G_0(x)=G(x,0)$ be the Green's function of the simple symmetric random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether
$$
\sum_{\substack{y\...
4
votes
1
answer
384
views
When is the Kochen-Stone inequality an equality?
The Kochen-Stone theorem says that if $A_n$ is sequence of events with $\sum_{i=1}^{\infty} P(A_i) = \infty$, then:
$$
P(A_n \mbox{ i.o.}) \ge \limsup_{n \to \infty} \frac{(\sum_{i=1}^nP(A_i))^2}{\...
4
votes
1
answer
184
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Asymptotics of a quotient related to a simple random walk
Let $\lambda_0 < \lambda_1$ and $\lambda_0 \lambda_1 > 1$ (i.e. at least $\lambda_1 > 1$). Further, let $S_n$ denote a simple random walk with increment distribution $$ P(X = 0)= P(X= 1) = 1/...
4
votes
1
answer
191
views
Rates of convergence to Tracy-Widom?
$\renewcommand{\!}{\mathbf}
\renewcommand{\Ai}{\operatorname{Ai}}$
One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where
$$\mathbf A(x, y)=\...
4
votes
2
answers
501
views
Bounding an expectation involving i.i.d. standard Gaussians and Rademacher
I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound:
$$
\phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d}
\mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(...
4
votes
1
answer
262
views
Berry–Esseen bound for operator norm of matrix averages
Is there a Berry–Esseen bound for operator norm of an average of independent random matrices?
Suppose $A_1, \dotsc, A_n$ are independent matrices with $\mathbb{E}[A_i] = I$ (the identity matrix). Is ...
4
votes
2
answers
182
views
Minimal coupling
Let $\mu,\nu$ be probability measures defined on a common measure space $(\Omega,\mathcal F)$. A coupling of $\mu,\nu$ is a probability measure $\pi$ on $(\Omega^2,\mathcal F^2)$ with marginals $\mu,\...
4
votes
1
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254
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On the convergence of the ratio of order statistics of gaps induced by $n$ uniform points on $[0,1].$
In an MO question here @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables ...
4
votes
1
answer
406
views
Stochastic processes and continuity of expectation
Let $X$ be a stochastic process with a.s. continuous sample paths on $[0, 1]$ such that $\mathbb E [X_t]$ is finite for all $t \in [0, 1]$. Given any non null subset $Y$ of the probability space, ...
4
votes
1
answer
582
views
Martingales and intersection of random walks
Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
4
votes
1
answer
263
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Information for discovering an item-colour assignment in a combinatorial game
We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
4
votes
1
answer
322
views
Rank of a random sparse matrix with nonnegative reals
I believe this should be some standard result in random matrices theory, but my initial search failed to find a definitive answer.
The question is given a random sparse matrix $M\in\mathbb{R}^{n\...
4
votes
1
answer
146
views
Large Deviations for Self-Normalized Sums
I am trying to understand the main result (Theorem 1.1) in this paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables
$$
\frac{\sum X_i}{\sqrt{n}\sqrt{\sum ...
4
votes
1
answer
407
views
Proving Conditional Independence
Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, is continuous and possibly has $ \mathbb{R} $ as its support. The random variable, $Z$, could be vector valued, but continuous.
I ...
4
votes
1
answer
175
views
Inner product of sorted Gaussian vector
Suppose $X_1,\ldots,X_n$ are i.i.d. standard normal. I'm wondering how to analyze the following quantity:
$$\left|\frac{X_{(1)}X_{(n)}+X_{(2)}X_{(n-1)}+\cdots+X_{(n)}X_{(1)}}{n}\right|$$
where $X_{(1)}...
4
votes
1
answer
385
views
Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function
Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on
$$
\mathbb{E}_{ZZ'}\left[ \exp\...
4
votes
1
answer
482
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 ...