All Questions
Tagged with pr.probability co.combinatorics
802 questions
2
votes
1
answer
188
views
Colored balls and bins -- asymptotic behavior
Suppose I have $N$ bins and a set of balls with $m$ different colors, where there are $n_i$ balls of color $i$. I also have values $0 < p_i \leq 1$ for all colors $i$. I throw all $\sum_i n_i$ ...
- 4,049
2
votes
1
answer
132
views
Independent decomposition of coordinate distribution
Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\...
- 515
2
votes
1
answer
62
views
Average number of rows to fit all elements in a multiset of natural numbers
Consider a multiset $S=\{a_1,a_2,...,a_{2n}\}$ of natural numbers. There are $2n$ elements (not necessarily unique since $S$ is a multiset) in $S$. All elements of $S$ belong to a set of natural ...
- 25
2
votes
1
answer
272
views
A generalization of negative binomial distribution
Assume we have a set of $n$ balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...
- 61
2
votes
3
answers
498
views
High order central moments of a symmetric binomial variable
Consider a random variable $X\sim B(n,\frac 12)$. I'm trying to estimate the asymptotic behaviour of its central moments $E((X-\frac n2)^r)$, where $r$ is even and in the range $\Omega(1)\leq r\leq O(...
2
votes
1
answer
363
views
Random walks with exponential decreasing steps
Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking $g^{...
- 1,648
2
votes
1
answer
723
views
Probability generating function zero implies random variable is infinite
Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
- 191
2
votes
1
answer
380
views
Parity, Balls and Boxes
Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...
- 263
2
votes
1
answer
115
views
Randomly chosen walk of fixed length
Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$.
A walk of ...
- 138
2
votes
1
answer
165
views
Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
- 153
2
votes
1
answer
118
views
Calculate the discrete probability of x number of good outcomes occurring before y number of bad outcomes
I have a grid of 16 tiles face down. Half are good outcomes and half are bad outcomes. How would I calculate the probability of picking x number of Good outcomes before y number of bad outcomes are ...
- 23
2
votes
1
answer
843
views
Interpretation of probability statements in Nina Zubrilina's paper
I asked this question on Math.stackexchange but got no answer.
In the paper Zubrilina - Asymptotic behavior of the edge metric dimension of the random graph (MR, the main result is
$$\operatorname{...
- 41
2
votes
2
answers
128
views
Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
- 128k
2
votes
1
answer
90
views
Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why?
This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
user125648
2
votes
1
answer
266
views
A question about finite free convolution
For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
- 2,246
2
votes
1
answer
112
views
Uniform bound for rare events
Consider a family $F$ of subsets of a probability space $\Omega$. Assume that $F$ has bounded VC dimension and that the measure of each subset in $F$ is at least $\epsilon$.
Drawing $n$ iid points ...
- 121
2
votes
1
answer
150
views
Probability of collision of some family of hash functions
Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
- 321
2
votes
1
answer
174
views
How to choose a random proper coloring
I am studying proper colorings of complete bipartite graphs and I'd like to be able to pick a random proper coloring and the compute some things about it.
Recall that a proper coloring of a complete ...
- 175
2
votes
1
answer
312
views
Probability of a contiguous sub-sequence with different elements
Let $a$ and $b$ be two positive integers, and say $b\gg a$. Let $S$ be a random sequence with $ab$ elements, whose entries are all integers from $1$ to $a$, such that each number from $1$ to $a$ ...
- 121
2
votes
1
answer
134
views
Completion time of a process on a tree
Given is a constant degree rooted tree of depth $D$. It is also known that the total number of nodes in the tree is at most $D^2$. There is a probabilistic process with discrete time steps on the ...
- 501
2
votes
1
answer
635
views
Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
- 201
2
votes
1
answer
467
views
Distribution on permutations derived from probability of pairwise orderings
A followup question to Probability estimates for pairwise majority votes - I think it doesn't actually give an answer in any terribly precise sense, but it would give something I'd be happy to use in ...
- 1,331
2
votes
0
answers
94
views
Concentration inequalities for functions of random binary strings
Let $(X_1,\ldots,X_n)$ be a vector in $\{0,1\}^n$ drawn uniformly at random among all vectors with exactly $k$ $1'$s. I am interested in inequalities for tail probabilities for the random variables $X,...
- 2,288
2
votes
0
answers
109
views
The fluctuations of a random path
Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
- 1,054
2
votes
0
answers
71
views
Distance distribution for Cayley graphs of the fintie Heisenberg groups H3(Z/nZ) approaches Gaussian for large "n"?
I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ).
Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? ...
- 24.9k
2
votes
0
answers
56
views
Dirichlet series solution to Poisson Point Process question (repost from math.SE)
Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
- 121
2
votes
0
answers
68
views
What is an efficient non-adaptive group testing scheme if the number of defectives, $d$, grows proportionally to the number of items, $n$?
Suppose that for some $p \in \left(0, 1\right)$ and some $n \in \mathbb{N}$, we have $n$ independent Bernoulli random variables, $X_{1}, X_{2}, \dots, X_{n}$, each with mean $p$. We shall call $X_{1}, ...
2
votes
0
answers
148
views
Union of two copies of uniform spanning forest on $\mathbb{Z}^3$ is transient? [closed]
Let $G$ be the (random) graph which is the union of two independent copies of the uniform spanning forest on $\mathbb{Z}^3$.
Question: Is (the simple random walk on) $G$ transient almost surely?
- 41
2
votes
0
answers
103
views
Optimization problem on randomly selecting subintervals from a given interval with combinatorial constraints
We select uniformly at random $k$ pairwise disjoint intervals from a given interval $[0,s]$ with length respectively equal to $\ell_1, \ell_2, \ldots, \ell_k\ $, i.e., we select uniformly at random $k$...
- 1,677
2
votes
0
answers
138
views
Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra
"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
- 10.5k
2
votes
1
answer
199
views
Do enough permutations of an initial set probably cover most permutations?
Fix $\alpha, \epsilon \in(0,1)$. Take $(S_n)_n$ to be any sequence of sets with each $S_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(...
2
votes
0
answers
164
views
Finding an optimal strategy for a combinatorial sequential game
We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
- 83
2
votes
0
answers
65
views
Are stable matchings (noise-)stable?
Suppose a group of computer scientists have entrusted their dating lives to a computer. Specifically, there are $n$ men and $n$ women, all of whom are cis-het. Being educated people, they of course ...
- 526
2
votes
0
answers
90
views
Motivation for proof of local lemma/construction version
I am interested in finding intuition to the bounds and proof of the asymmetric local lemma.
I think the $k$-SAT is fairly intuitive, but I would like to understand the general version.
One good ...
- 515
2
votes
0
answers
165
views
Ask for some reference about isoperimetric constant on Voronoi diagrams?
Given a Poisson point process $\mathcal{P}$ in $\mathbb{R}^2$, the $\textbf{Voronoi cells}$ of a point $p\in \mathcal{P}$ is defined by
$$V(x):=\{y\in \mathbb{R}^2: \|x-y\|=\min_{x'\in \mathcal{P}}\|x'...
- 288
2
votes
0
answers
80
views
Small set in partition-large class
A collection $\mathcal{A}\subseteq \mathcal{P}(X)$
is $k$-large in $X$
if for every $k$-partition
of $X$ namely
$X_1,\cdots,X_k$, there exists an $i\leq k$ such that $X_i\in \mathcal{A}$;
$\mathcal{...
- 909
2
votes
0
answers
85
views
How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?
Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...
- 267
2
votes
0
answers
83
views
Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
- 13.9k
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
2
votes
0
answers
58
views
Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$
Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
- 6,853
2
votes
0
answers
78
views
Width of symmetric groups
MSE crosspost
For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...
- 4,600
2
votes
1
answer
508
views
Proof and interpretation of the following percolation theory result for $n\times n$ square grid
While I was discussing this question with @JamesMartin, he mentioned a result here that:
In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such
that $\epsilon>0$ and $p_c$ is the ...
user123818
2
votes
0
answers
59
views
Min/max row-sum distribution of a symmetric matrix of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then ...
- 272
2
votes
0
answers
91
views
Semigroups of nondecreasing functions
Consider some partially ordered set $(E,\leq)$. Assume either that it is countable with the discrete topology, or that it has some topology compatible with the order, preferably one that makes it into ...
2
votes
0
answers
59
views
Totally distance non-preserving transformations
JL lemma (https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma)
guarantees if you have a set of $K$ points in $\Bbb R^N$ a random transformation guarantees that the set can be projected ...
- 13.9k
2
votes
0
answers
93
views
Erdös-Renyi Model with prescribed subgraph
In the Erdös-Rényi model for random graphs there is a lot of results stating sharp phase transitions for the probability of a random graph to contain a fixed prescribed ...
- 2,630
2
votes
0
answers
227
views
Average minimum number of random k-sparse vectors in $\mathbb{F}_2^n$ to span a specific base vector?
A while back I posted a question in MO about the average minimum number of independent random k-sparse (having at most $k$ non-zero elements) vectors belonging to $\mathbb{F}_2^n$ to span the whole ...
- 307
2
votes
0
answers
386
views
A two variable recurrence relation with conditionals
I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence
$$
f(n,m) = \begin{cases} f(n, \...
- 305
2
votes
0
answers
72
views
Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$
Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
- 305
2
votes
0
answers
116
views
Number of self avoiding paths which are not ``tie together''
Consider the lattice $\mathbb{Z}^d$. Let $A_{n}$ be the set of returning self-avoiding paths (from $0$ to $0$) having length $n$. For any path $\omega \in A_{n}$, let $f(\omega)$ be the number of ...
- 295