Ballot Theorem Research Articles

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions, stays negative as the ratio of the radii of the outer and the inner boundary tends to infinity. Such estimates are often needed in the study of extreme values of the discrete Gaussian free field on planar domains.

SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM

We present a recursive formula for the two-sided ballot theorem using left and right shift transforms. In particular, we showed that the xth entry of the image of the d + 1 dimensional unit vector under the sum of the left and right shift operators is the number of walks in the lattice interval [0,d] that start at the origin and stop at the location x. This approach enables us to write a recursive formula for the number of possible n−walks between two obstacles that stop at a predetermined location.

Unusually large components in near-critical Erdős–Rényi graphs via ballot theorems

AbstractWe consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and A is large, \begin{equation*}\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}},\end{equation*} where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows A and $\lambda$ to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.

Discrete-time TASEP with holdback

We study the following interacting particle system. There are ρn particles, ρ<1, moving clockwise (“right”), in discrete time, on n sites arranged in a circle. Each site may contain at most one particle. At each time, a particle may move to the right-neighbor site according to the following rules. If its right-neighbor site is occupied by another particle, the particle does not move. If the particle has unoccupied sites (“holes”) as neighbors on both sides, it moves right with probability 1. If the particle has a hole as the right-neighbor and an occupied site as the left-neighbor, it moves right with probability 0<p<1. (We refer to the latter rule as a “holdback” property.) From the point of view of holes moving counter-clockwise, this is a zero-range process.The main question we address is: what is the system steady-state flux (or throughput) when n is large, as a function of density ρ? The most interesting range of densities is 0≤ρ<1∕2. We define the system typical flux as the limit in n→∞ of the steady-state flux in a system subject to additional random perturbations, when the perturbation rate vanishes. Our main results show that: (a) the typical flux is different from the formal flux, defined as the limit in n→∞ of the steady-state flux in the system without perturbations, and (b) there is a phase transition at density h=p∕(1+p). If ρ<h, the typical flux is equal to ρ, which coincides with the formal flux. If ρ>h, a condensation phenomenon occurs, namely the formation and persistence of large particle clusters; in particular, the typical flux in this case is p(1−ρ)<h<ρ, which differs from the formal flux when h<ρ<1∕2.Our results include both the steady-state analysis (which determines the typical flux) and the transient analysis. In particular, we derive a version of the Ballot Theorem, and show that the key “reason” for large cluster formation for densities ρ>h is described by this theorem.

Modelling the order of scoring in team sports

Abstract This paper considers sports matches in which two teams compete to score more points within a set amount of time (e.g. football, ice hockey). We focus on the order in which the competing teams score during the match (order of scoring). This type of order of scoring problem has not been addressed previously, and doing so here gives new insights into sports matches. For example, our analysis can deal with a situation that spectators find matches that involve comebacks particularly exciting. To describe such problems mathematically, we formulate the probabilities of (i) the favourite team leading throughout the match and (ii) the favourite team falling behind the opposing team but then making a comeback. These probabilities are derived using an independent Poisson model and lattice path enumeration, the latter of which involves the well-known ballot theorem. The independence assumption allows lattice path enumeration to be applied directly to the Poisson model and various scoring patterns to be addressed. We confirm that the values obtained from the proposed models agree well with actual sports data from football, futsal and ice hockey.

A cyclic approach on classical ruin model

The ruin problem has long since received much attention in the literature. Under the classical compound Poisson risk model, elegant results have been obtained in the past few decades. We revisit the finite-time ruin probability by using the idea of cycle lemma, which was used in proving the ballot theorem. The finite-time result is then extended to infinite-time horizon by applying the weak law of large numbers. The cycle lemma also motivates us to study the claim instants retrospectively, and this idea can be used to reach the ladder height distribution on the infinite-time horizon. The new proofs in this paper link the classical finite-time and infinite-time ruin results, and give an intuitive way to understand the nature of ruin.

Short proofs in extrema of spectrally one sided Lévy processes

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Levy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Levy processes. An analogous formula is obtained for the supremum of spectrally negative Levy processes.

Bertrand’s Ballot Theorem

Summary In this article we formalize the Bertrand’s Ballot Theorem based on [17]. Suppose that in an election we have two candidates: A that receives n votes and B that receives k votes, and additionally n ≥ k. Then this theorem states that the probability of the situation where A maintains more votes than B throughout the counting of the ballots is equal to (n − k)/(n + k). This theorem is item #30 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Analysis of Buffer Starvation With Application to Objective QoE Optimization of Streaming Services

Our purpose in this paper is to characterize buffer starvations for streaming services. The buffer is modeled as an M/M/1 queue, plus the consideration of bursty arrivals. When the buffer is empty, the service restarts after a certain amount of packets are \emph{prefetched}. With this goal, we propose two approaches to obtain the \emph{exact distribution} of the number of buffer starvations, one of which is based on \emph{Ballot theorem}, and the other uses recursive equations. The Ballot theorem approach gives an explicit result. We extend this approach to the scenario with a constant playback rate using T\`{a}kacs Ballot theorem. The recursive approach, though not offering an explicit result, can obtain the distribution of starvations with non-independent and identically distributed (i.i.d.) arrival process in which an ON/OFF bursty arrival process is considered in this work. We further compute the starvation probability as a function of the amount of prefetched packets for a large number of files via a fluid analysis. Among many potential applications of starvation analysis, we show how to apply it to optimize the objective quality of experience (QoE) of media streaming, by exploiting the tradeoff between startup/rebuffering delay and starvations.

Coding multitype forests: Application to the law of the total population of branching forests

By extending the breadth first search algorithm to any $d$-type critical or subcritical irreducible branching tree, we show that such trees may be encoded through $d$ independent, integer valued, $d$-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process.

A Note on Wiener–Hopf Factorization for Markov Additive Processes

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.

The Hitting Time Theorem Revisited

This note presents a very short and simple combinatorial proof of the classical hitting time theorem, or equivalently, the ballot theorem.

Direct consequences of the basic Ballot Theorem

We use only the classic basic ballot result and simple combinatorial arguments to derive the distributions of the first passage time and the number of visits in the usual random walk model.

The structure of the allelic partition of the total population for Galton–Watson processes with neutral mutations

We consider a (sub-)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.

Application of the generalised ballot theorem for evaluation of performance in packet buffers with non-first in first out scheduling

Packet scheduling is a vital component to support different classes of service in all-packet networks. In classical queuing systems, the waiting-time performance of non-first in first out buffer scheduling systems could be predicted through the use of analysis. However, all-packet networks feature traffic patterns that do not conform to classical Poisson-like processes, and this greatly complicates the evaluation of their performance. Our novel approach to this problem is through a hybrid combination of analysis and simulation. The authors derive a combinatorial algorithm, using the generalised ballot theorem, which predicts waiting times for low-priority traffic. When this algorithm is combined with prior work on traffic aggregation, the authors achieve a significant reduction in the state space associated with the buffer under study. To numerically test this algorithm, the authors demonstrate its use in simulation, where state space and event count reduction is a fundamental requirement to ensure experiments complete in a timely fashion. Numerical results from these simulations show a very significant reduction in the number of events processed combined with improved state coverage. This is achieved while maintaining a highly accurate representation of packet delays compared with a conventional approach.

The number of lattice paths below a cyclically shifting boundary

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.”

Four Proofs of the Ballot Theorem

(2007). Four Proofs of the Ballot Theorem. Mathematics Magazine: Vol. 80, No. 5, pp. 345-352.

On finite-time ruin probabilities for classical risk models

This paper examines the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard & Lefèvre (1997) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.

The distribution of the number of arrivals in a subinterval of a busy period of a single server queue

In the course of attempting to estimate the arrival rate of a single server queue using an active probing experiment, the authors found it necessary to derive the distribution of the number of arrivals between two probes under the conditions that the busy period of the queue lasts this long. In this paper we derive this distribution. The key building blocks in the derivation of the distribution are the classical ballot theorem and its generalized forms.

Quasi-Birth-and-Death Processes with an Explicit Rate Matrix

We determine the equilibrium distribution for a class of quasi-birth-and-death (QBD) processes using the matrix-geometric method, which requires the determination of the rate matrix R. In contrast to most QBD processes, the class under consideration allows for an explicit description of R, by exploiting its probabilistic interpretation. We show that the problem of finding each element of R reduces to counting lattice paths in the transition diagram. For resolving this counting problem, we present an extended version of the classical Ballot theorem. We also present various examples of queueing models that fit into the class.