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Svante Janson

Publications and source records attributed to Svante Janson.

At least 91 records · Page 5Linked to original sources
arXiv2018

Tensor norms on ordered normed spaces, polarization constants, and exchangeable distributions

We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The results are applied to the problem of representing a finitely exchangeable distribution as a mixture of powers, i.e, mixture of distributions of i.i.d. sequences, using a signed mixing measure.

math.FA↗
arXiv2018

Phragmén's and Thiele's election methods

The election methods introduced in 1894--1895 by Phragmén and Thiele, and their somewhat later versions for ordered (ranked) ballots, are discussed in detail. The paper includes definitions and examples and discussion of whether the methods satisfy some properties, including monotonicity, consistency and various proportionality criteria. The relation with STV is also discussed. The paper also contains historical information on the methods.

math.HO↗
arXiv2018

Thresholds quantifying proportionality criteria for election methods

We define several different thresholds for election methods by considering different scenarios, corresponding to different proportionality criteria that have been proposed by various authors. In particular, we reformulate the criteria known as DPC, PSC, JR, PJR, EJR in our setting. We consider multi-winner election methods of different types, using ballots with unordered lists of candidates or ordered lists, and for comparison also methods using only party lists. The thresholds are calculated for many different election methods. The considered methods include classical ones such as BV, SNTV and STV (with some results going back to e.g. Droop and Dodgson in the 19th century); we also study in detail several perhaps lesser known methods by Phragmén and Thiele. There are also many cases left as open problems.

cs.GT↗
arXiv2018

Long term behaviour of a reversible system of interacting random walks

This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of alternative proofs not requiring reversibility are provided.

math.PR↗
arXiv2018

The hiring problem with rank-based strategies

The hiring problem is studied for general strategies based only on the relative ranking of the candidates; this includes some well known strategies studied before such as hiring above the median. We give general limit theorems for the number of hired candidates and some other properties, extending previous results. The results exhibit a dichotomy between two classes of rank-based strategies: either the asymptotics of the process are determined by the early events, with a.s. convergence of suitably normalized random variables, or there is a mixing behaviour without long-term memory and with asymptotic normality.

math.PR↗
arXiv2018

Inversions in split trees and conditional Galton--Watson trees

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the $k$-total common ancestors of $T$ (an extension of the total path length). Then we consider $X_n$, the normalized version of $I(T_n)$, for a sequence of trees $T_n$. For fixed $T_{n}$'s, we prove a sufficient condition for $X_n$ to converge in distribution. As an application, we identify the limit of $X_n$ for complete $b$-ary trees. For $T_n$ being split trees, we show that $X_n$ converges to the unique solution of a distributional equation. Finally, when $T_n$'s are conditional Galton--Watson trees, we show that $X_n$ converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that are stronger and much broader compared to previous work by Panholzer and Seitz.

math.PR↗
arXiv2018

Asymptotic normality in Crump-Mode-Jagers processes: the lattice case

Consider a supercritical Crump--Mode--Jagers process such that all births are at integer times (the lattice case). We show that under a certain condition on the intensity of the offspring process, the second-order fluctuations of the age distribution are asymptotically normal; the condition is essential and not just a technicality. This extends to populations counted by a random characteristic.

math.PR↗
arXiv2018

Asymptotics of fluctuations in Crump-Mode-Jagers processes: the lattice case

Consider a supercritical Crump--Mode--Jagers process such that all births are at integer times (the lattice case). Let $\widehatμ(z)$ be the generating function of the intensity of the offspring process, and consider the complex roots of $\widehatμ(z)=1$. The smallest (in absolute value) such root is $e^{-α}$, where $α>0$ is the Malthusian parameter; let $γ_*$ be the second smallest absolute value of a root. We show, assuming some technical conditions, that there are three cases: (i) if $γ_*>e^{-α/2}$, then the second-order fluctuations of the age distribution are asymptotically normal; (ii) if $γ_*=e^{-α/2}$, then the fluctuations are still asymptotically normal, but with a larger order of the variance; (iii) if $γ_*<e^{-α/2}$, then the fluctuations are even larger, but will oscillate and (except in degenerate cases) not converge in distribution. This trichotomy is similar to what has been seen in related situations, e.g. for some other branching processes, and for Pólya urns. The results lead to a symbolic calculus describing the limits. The results extends to populations counted by a random characteristic.

math.PR↗
arXiv2018

Preferential Attachment When Stable

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $α^{th}$ power $(α>1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, a fine control on this probability may be leveraged to derive a lower tail Large Deviation Principle (LDP) for $L = \sum_{i=1}^{n} \frac{S_i^2}{i^2}$, where $\{S_n : n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternate proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous time analogue of $L$. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

math.PR↗
arXiv2018

Patterns in random permutations avoiding some sets of multiple patterns

We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $σ$ has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.

math.PR↗
arXiv2018

Renewal theory for asymmetric $U$-statistics

We extend a functional limit theorem for symmetric $U$-statistics [Miller and Sen, 1972] to asymmetric $U$-statistics, and use this to show some renewal theory results for asymmetric $U$-statistics. Some applications are given.

math.PR↗
arXiv2018

A.s. convergence for infinite colour Pólya urns associated with random walks

We consider Pólya urns with infinitely many colours that are of a random walk type, in two related version. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014--2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).

math.PR↗
arXiv2018

Non-fringe subtrees in conditioned Galton--Watson trees

We study $S(\mathcal T_{n})$, the number of subtrees in a conditioned Galton--Watson tree of size $n$. With two very different methods, we show that $\log(S(\mathcal T_{n}))$ has a Central Limit Law and that the moments of $S(\mathcal T_{n})$ are of exponential scale.

math.CO↗
arXiv2017

Patterns in random permutations avoiding the pattern 321

We consider a random permutation drawn from the set of 321-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $σ$ has a limit distribution, after scaling by $n^{m+\ell}$ where $m$ is the length of $σ$ and $\ell$ is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

math.PR↗
arXiv2017

Random replacements in Pólya urns with infinitely many colours

We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times[0,1]$.

math.PR↗
arXiv2017

Competing first passage percolation on random graphs with finite variance degrees

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate $λ_1$ ($λ_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $λ_1=λ_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $λ_1\neq λ_2$, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

math.PR↗
arXiv2017

Competition in growth and urns

We study survival among two competing types in two settings: a planar growth model related to two-neighbour bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncoloured sites are given a colour at rate $0$, $1$ or $\infty$, depending on whether they have zero, one, or at least two neighbours of that colour. In the urn scheme, each vertex of a graph $G$ has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same colour is added to each neighbouring urn, and balls in the same urn but of different colours annihilate on a one-for-one basis. We show that, for every connected graph $G$ and every initial configuration, only one colour survives almost surely. As a corollary, we deduce that in the two-type growth model on $\mathbb{Z}^2$, one of the colours only infects a finite number of sites with probability one. We also discuss generalisations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

math.PR↗