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.
9,024 questions
0
votes
0
answers
337
views
What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?
This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
CommunityBot
- 1
2
votes
2
answers
9k
views
Multiplying two probability distributions represented by particles
Hello,
I have two probability distributions p1(x) and p2(x) given by (x_1i, w_1i) and (x_2i, w_2i) respectively, i.e. they are both represented by sets of particles. I need to create the pdf p(x) = ...
4
votes
2
answers
882
views
Distribution of a maximum
I am reposting a question on math.stackexchange which did not recieve good questions.
The orginal questio is at https://math.stackexchange.com/questions/73091/distribution-of-a-maximum.
Randomly ...
CommunityBot
- 1
5
votes
1
answer
183
views
Stable Law with Rates
If $X_{i}$ are a bunch of iid random variables with mean 0 and finite second moments, we know that $\sum_{i=1}^{n} \frac{X_{i}}{\sqrt{n}}$ converges in law to a Gaussian. Furthermore, by the Berry-...
- 83
8
votes
1
answer
1k
views
Expected norm of sum of random orthogonal matrices
Somehow I got wondering about the following question today:
Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices.
What is the expected value of the quantity $\|\...
- 28.6k
16
votes
1
answer
276
views
Length of the last edge when visiting points by nearest neighbor order
Take $n$ points uniformly in $[0,1] \times [0,1]$. Then pick uniformly $X_0$ one of these points as your starting point. Then let $X_1$ be the nearest neighbor of $X_0$, let $X_2$ be the nearest ...
3
votes
1
answer
294
views
Convergence of stopped Brownian motion
Suppose $B$, $B_n$ are Brownian motions, and write $B^s$ for $B$ stopped at the first time equals $k$, say. (Similarly $B^s_n$).
I know how to prove the following: if $B_n \to B$ uniformly on ...
1
vote
1
answer
742
views
proofs of stochastic boundedness
I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to ...
- 6,711
4
votes
5
answers
492
views
Some questions concerning a random number process
Consider the following Markov process: Start with an integer $N = N_0$. Now repeatedly choose an $N_i$ uniformly at random in the range $[1...N_{i-1}]$ until $N_i = 1$ at which point one terminates ...
CommunityBot
- 1
4
votes
2
answers
350
views
analogue of GUE and Ginibre in higher dimensions
This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution:
$$\displaystyle p(z_1,\ldots, z_N) = C_N \exp\left(-\...
- 2,135
17
votes
1
answer
9k
views
Intuitive understanding of the Stieltjes transform
I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance ...
CommunityBot
- 1
2
votes
1
answer
395
views
Probability-one event for Markov chain
Let $X$ be a Markov chain, with countable state space $I$ and transition probability matrix $P$. $X$ is irreducible, but need not be recurrent. Let $S$ be a fixed subset of $I$.
Define a subset $K$ ...
CommunityBot
- 1
6
votes
1
answer
1k
views
Low degree polynomial approximation for the entropy function
Let $X$ be a discrete random variable with possible values
$\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of
$X$. In addition, denote $p_i=p(x_i)$.
The entropy of $X$ is ...
- 188k
2
votes
1
answer
262
views
Processes approximating a reflected brownian motion.
Let $W$ be a standard Brownian Motion. Let $\epsilon>0$ be given. Let $X^\epsilon$ be the process which diffuses like $W$ on $(-\epsilon,\infty)$, but when $X^\epsilon$ reaches the level $-\...
- 3,069
10
votes
1
answer
462
views
For what range of edge probability does the following property hold for random graphs?
Let $G(n,p)$ denote the Erdős–Rényi model of random graph. For a given function $p = p(n)$ we say that $G \in G(n,p)$ asymptotically almost surely has property $\mathcal{P}$ if
$$\mbox{Pr}[G \mbox{ ...
- 6,711
3
votes
1
answer
242
views
Bounding the success time of a coupon collector like problem
Consider the complete graph on $n$ vertices. Each step, one chooses one of the $\binom{n}{2}$ edges iid uniformly at random. Say a sequence of choice is successful if there is some permutation of the ...
- 23.2k
0
votes
1
answer
426
views
Lower bounds for partial sums of multiplicative functions
The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\...
- 12.8k
7
votes
3
answers
415
views
Markov Property: determined by just the law or also the realization?
When one says that a stochastic process is Markovian, is this a property solely of the law of the process, or does the realization of the process also come in to play? I am asking even for the ...
CommunityBot
- 1
0
votes
2
answers
327
views
Copulas and time series
Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?
- 1,334
3
votes
0
answers
211
views
Elementary analysis: reference request
Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...
- 2,895
1
vote
1
answer
502
views
nonnegative series expansion of nonnegative functions
The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
CommunityBot
- 1
0
votes
1
answer
377
views
Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
CommunityBot
- 1
3
votes
0
answers
171
views
Iterated Kumaraswamy distributions
The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$.
Does anyone know any formulas or properties relating to iterations of this on itself, meaning
$$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$
If ...
- 233
10
votes
2
answers
3k
views
Statistics for Haar measure of random matrices?
Let's say I have $M$ samples of $N\times N$ real orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over $O(N)$?...
CommunityBot
- 1
1
vote
0
answers
111
views
Conditional prob of value given input is drawn from subset = conditioning over subset?
Hey guys, I have a pretty basic question that I want to be sure of. I'm taking a probability over an input selected uniformly at random from binary strings of length $l(n)$. I would like to compare ...
- 113
3
votes
2
answers
1k
views
Sequences of linear combinations of measures
Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
- 60.6k
1
vote
1
answer
301
views
Distances between and among points in a region
Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - ...
- 32.4k
10
votes
0
answers
533
views
Abelian sandpile models
This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as ...
CommunityBot
- 1
-3
votes
1
answer
332
views
Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
0
votes
0
answers
479
views
Passage Time Distributions for Poisson processes.
Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. ...
- 943
6
votes
1
answer
3k
views
expectation of supremum
Hello,
Suppose $(X_{n}(t))_{n\geq 1}$ is a sequence of real valued stochastic processes, and $T>0$ a fixed number.
Do we have the following implication ?
$\displaystyle{ \lim_{n \to \infty} \...
- 840
4
votes
2
answers
295
views
Distribution of the biggest gap
Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a_1,a_2,\dots,a_n\)}$, and then we can ...
CommunityBot
- 1
6
votes
0
answers
1k
views
Relationship between R-transform and free convolution of random matrices?
I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
- 1,890
1
vote
0
answers
215
views
Has this process been studied?
Take a Poisson process on $\mathbb{R}$ with intensity given by Lebesgue measure. Think of this as the measure $d\mu=\sum_{n} \delta(t-\xi_n )dt$ where $\xi_n$ are the points of the process. Now ...
- 1,471
1
vote
0
answers
228
views
Bounding a stochastic process in terms of time to return to 0
I have a $\mathbb{Z}_+$-valued stochastic process $X$ in discrete time, which has unit jumps up or down. I know the following statement: there exists a random variable $\tau$, almost surely finite, s....
- 1,069
2
votes
0
answers
198
views
Forcing the existence of a Condorcet Winner
Suppose that there is an election with three candidate and an infinite number of voters whose opinion lie in a two-dimensional issue space according to some distribution, and that voter's candidate ...
- 457
10
votes
3
answers
4k
views
Random bipartite graphs
Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I ...
- 37.7k
5
votes
1
answer
225
views
Subadditive Kingmans theorem for lattices.
I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need.
I would rather have something like that:
Let us consider $\mathbb{Z}^...
CommunityBot
- 1
1
vote
2
answers
837
views
A difficult (?) multinomial problem (balls, bins, etc.)
Consider the well known multinomial setting: there are L balls, thrown at random at n bins so that the probability that a ball falls in bin i is $p_i$, independent of the other balls (the $p_i$’s are ...
- 5,492
1
vote
0
answers
135
views
Optimizing for a unique outcome of a probabilistic marriage problem
Let's say I have some number of individuals who are single, $(b_1, ..., b_N) \in B$, and for every possible pairing of two individuals, $b_i$ and $b_j$, I happen to know the exact probability that the ...
- 11
1
vote
1
answer
332
views
Product of a transient and a positive recurrent Markov chain
Let $X$ be a transient Markov chain with countable state space $S(X)$. Let $Y$ be a positive recurrent Markov chain with countable state space $S(Y)$. (Time is discrete.)
Let $A \subseteq S(X)$ be ...
- 5,721
1
vote
2
answers
1k
views
Tail Conditional Expectation of a binomial random variable
Let $X \sim B(n,c/n)$ be a binomially distributed random variable with
parameter $p = c/n$, and hence mean $c$. Here $c$ is some function of $n$ such that
i) $c \geq n^{2/3}$
ii) The function $c$ ...
- 37.7k
12
votes
3
answers
1k
views
Is there a simple inductive procedure for generating labeled trees uniformly at random, without direct recourse to Prüfer sequences?
Suppose you have a labeled tree $T$ on vertices $V=\lbrace 1,\ldots,n\rbrace$ that is drawn uniformly at random from the set of all $n^{n-2}$ such trees. I am seeking an $f$ satisfying the following ...
- 37.7k
3
votes
1
answer
751
views
Will a given pattern ever show up in an infinite random sequence of 0s and 1s?
Here the pattern is a finite or infinite sequence of 0s and 1s, not necessarily consecutive, for example, $\lbrace 1, *, 1, *, 1 \rbrace$ and $\lbrace 0, *, 0, *, 0, *, \ldots \rbrace$ ($ * $, hole ...
- 33
0
votes
4
answers
386
views
Recovering a function from a set of approximations
We assume that we have a finite set of agents with approximate knowledge about a certain function, and from this collection of approximations we want to recover the actual value of the function.
More ...
CommunityBot
- 1
5
votes
0
answers
227
views
Number of times lead changes in a multi-candidate election (reference-request)
In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
- 51
0
votes
1
answer
3k
views
Conditional expectation of a product
I have an expression: $E[(b+X)^2|Y]$ where $X$ and $Y$ are normally distributed random variables, being two components of a final unknown outcome $Z$ ($Y$ is known, $X$ is the noise component):
$Y$ =...
- 256
1
vote
1
answer
259
views
Amenability with respect to a function
Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $...
CommunityBot
- 1
0
votes
1
answer
774
views
A question on independence
For each natural number $n \geq 2$, define the set $A_n$ to be the set of points $p/n$ with $0 < p < n, \gcd(p,n) = 1$. Now define a sequence of independent random variables $X_1, X_2, \cdots$, ...
CommunityBot
- 1
7
votes
4
answers
4k
views
A formal definition of Scaling Limits?
I'm looking for a formal definition of scaling limit in a rigorous math sense, also, if somebody knows a good translation to spanish. A good bibliography could be helpful.
- 3,069