scholarly journals First Occurrence in Pairs of Long Words: A Penney-ante Conjecture of Pevzner

1995 ◽  
Vol 4 (3) ◽  
pp. 279-285
Author(s):  
Dudley Stark
Keyword(s):  
Probability Distribution ◽  
Unique Element ◽  
Secondary Maximum ◽  
Maximum Value ◽  
Random Elements ◽  
Finite Set ◽  
First Occurrence ◽  
Independent And Identically Distributed

Suppose X1, X2,… is a sequence of independent and identically distributed random elements whose values are taken in a finite set S of size |S| ≥ 2 with probability distribution ℙ(X = s) = p(s) > 0 for s ∈ S. Pevzner has conjectured that for every probability distribution ℙ there exists an N > 0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1, X2,… has greater probability than that of A appearing before B. In this paper it is shown that a distribution ℙ satisfies Pevzner's conclusion if and only if the maximum value of ℙ, p, and the secondary maximum c satisfy the inequality . For |S| = 2 or |S| = 3, the inequality is true and the conjecture holds. If , then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mode.

Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

scholarly journals Posterior Probability on Finite Set

2012 ◽  
Vol 20 (4) ◽  
pp. 257-263
Author(s):  
Hiroyuki Okazaki
Keyword(s):  
Probability Distribution ◽  
Probability Measure ◽  
Posterior Probability ◽  
Probability Distributions ◽  
Sample Space ◽  
Finite Sample ◽  
Finite Set

Summary In [14] we formalized probability and probability distribution on a finite sample space. In this article first we propose a formalization of the class of finite sample spaces whose element’s probability distributions are equivalent with each other. Next, we formalize the probability measure of the class of sample spaces we have formalized above. Finally, we formalize the sampling and posterior probability.

On random mappings with a single attracting centre

1987 ◽  
Vol 24 (1) ◽  
pp. 258-264 ◽  
Author(s):  
Ljuben R. Mutafchiev
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Vector ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Random Mappings ◽  
Vertex Set ◽  
Independent Identically Distributed ◽  
Independent And Identically Distributed

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = Pj, j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph GT with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of GT into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P0. It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.

Distribution of the Time of the First k-Record

1997 ◽  
Vol 11 (3) ◽  
pp. 273-278 ◽  
Author(s):  
Ilan Adler ◽  
Sheldon M. Ross
Keyword(s):  
Generating Function ◽  
Random Variables ◽  
Recursive Formula ◽  
Finite Set ◽  
Independent And Identically Distributed

We compute the first two moments and give a recursive formula for the generating function of the first k-record index for a sequence of independent and identically distributed random variables that take on a finite set of possible values. When the random variables have an infinite support, we bound the distribution of the index of the first k-record and show that its mean is infinite.

scholarly journals The Generalised Coupon Collector Problem

2008 ◽  
Vol 45 (03) ◽  
pp. 621-629 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Probability Distribution ◽  
Asymptotic Distribution ◽  
Birthday Problem ◽  
Coupon Collector ◽  
Poisson Limit ◽  
Special Case ◽  
Independent And Identically Distributed

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

New maximal inequalities for N-demimartingales with scan statistic applications

2017 ◽  
Vol 54 (2) ◽  
pp. 363-378 ◽  
Author(s):  
Markos V. Koutras ◽  
Demetrios P. Lyberopoulos
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Cumulative Distribution Function ◽  
Numerical Study ◽  
Cumulative Distribution ◽  
Scan Statistic ◽  
Maximal Inequalities ◽  
Binary Trials ◽  
First Occurrence ◽  
Independent And Identically Distributed

Abstract In the present work, some new maximal inequalities for nonnegative N-demi(super)martingales are first developed. As an application, new bounds for the cumulative distribution function of the waiting time for the first occurrence of a scan statistic in a sequence of independent and identically distributed (i.i.d.) binary trials are obtained. A numerical study is also carried out for investigating the behavior of the new bounds.

scholarly journals Generating Random Elements in Finite Groups

10.37236/818 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Finite Group ◽  
Probability Distribution ◽  
Finite Groups ◽  
Random Elements ◽  
Random Generators ◽  
A Finite Group

Let $G$ be a finite group of order $g$. A probability distribution $Z$ on $G\ $is called $\varepsilon$-uniform if $\left\vert Z(x)-1/g\right\vert \leq\varepsilon/g$ for each $x\in G$. If $x_{1},x_{2},\dots,x_{m}$ is a list of elements of $G$, then the random cube $Z_{m}:=Cube(x_{1},\dots,x_{m}) $ is the probability distribution where $Z_{m}(y)$ is proportional to the number of ways in which $y$ can be written as a product $x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\cdots x_{m}^{\varepsilon_{m}}$ with each $\varepsilon _{i}=0$ or $1$. Let $x_{1},\dots,x_{d} $ be a list of generators for $G$ and consider a sequence of cubes $W_{k}:=Cube(x_{k}^{-1},\dots,x_{1}^{-1},x_{1},\dots,x_{k})$ where, for $k>d$, $x_{k}$ is chosen at random from $W_{k-1}$. Then we prove that for each $\delta>0$ there is a constant $K_{\delta}>0$ independent of $G$ such that, with probability at least $1-\delta$, the distribution $W_{m}$ is $1/4$-uniform when $m\geq d+K_{\delta }\lg\left\vert G\right\vert $. This justifies a proposed algorithm of Gene Cooperman for constructing random generators for groups. We also consider modifications of this algorithm which may be more suitable in practice.

scholarly journals Unanimous and Strategy-Proof Probabilistic Rules for Single-Peaked Preference Profiles on Graphs

2021 ◽  
Author(s):  
Hans Peters ◽  
Souvik Roy ◽  
Soumyarup Sadhukhan
Keyword(s):  
Probability Distribution ◽  
Connected Graph ◽  
Line Graph ◽  
Preference Profile ◽  
Tree Representation ◽  
General Characterization ◽  
Finite Set ◽  
Probabilistic Rule ◽  
Strategy Proof ◽  
Block Tree

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

scholarly journals Strong T-Coloring of Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 4677-4681
Keyword(s):  
Chromatic Number ◽  
Maximum Value ◽  
Finite Set ◽  
Minimum Number ◽  
Edge Span ◽  
Positive Integers

A 𝑻-coloring of a graph 𝑮 = (𝑽,𝑬) is the generalized coloring of a graph. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set T of positive integers containing 𝟎 , a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. We define Strong 𝑻-coloring (S𝑻-coloring , in short), as a generalization of 𝑻-coloring as follows. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a S𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻 and |𝒇(𝒖) − 𝒇(𝒘)| ≠ |𝒇(𝒙) − 𝒇(𝒚)| for any two distinct edges 𝒖𝒘, 𝒙𝒚 in 𝑬(𝑮). The S𝑻-Chromatic number of 𝑮 is the minimum number of colors needed for a S𝑻-coloring of 𝑮 and it is denoted by 𝝌𝑺𝑻(𝑮) . For a S𝑻 coloring 𝒄 of a graph 𝑮 we define the 𝒄𝑺𝑻- span 𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all pairs 𝒖, 𝒗 of vertices of 𝑮 and the S𝑻 -span 𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒔𝒑𝑺𝑻(𝑮) = min 𝒔𝒑𝑺𝑻 𝒄 (𝑮) where the minimum is taken over all ST-coloring c of G. Similarly the 𝒄𝑺𝑻-edgespan 𝒆𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all edges 𝒖𝒗 of 𝑮 and the S𝑻-edge span 𝒆𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒆𝒔𝒑𝑺𝑻(𝑮) = min 𝒆𝒔𝒑𝑺𝑻 𝒄 𝑮 where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S𝑻- chromatic number, 𝒔𝒑𝑺𝑻(𝑮) , and 𝒆𝒔𝒑𝑺𝑻(𝑮) of graphs.

On random mappings with a single attracting centre

1987 ◽  
Vol 24 (01) ◽  
pp. 258-264 ◽  
Author(s):  
Ljuben R. Mutafchiev
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Vector ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Random Mappings ◽  
Vertex Set ◽  
Independent Identically Distributed ◽  
Independent And Identically Distributed

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = P j , j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph G T with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of G T into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P 0 . It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.

Sign in / Sign up

Export Citation Format

Share Document