Regularly varying functions and Karamata’s representation

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Regularly varying functions and Karamata’s representation

The characterization of the max-domain of attraction and the proof of Theorem 2.2 is based on the theory of regularly varying functions. In this subsection, we recall some basic notions, used in this thesis. Further information can be found in the following references: [BGT87, Fel71, dH70, Sen76]. A measurable function U : R+ ! R+ is said to be -regularly varying at innity (respectively at zero) if lim t!1 U(tx) U(t) = x; for every x > 0; in the case of regularly variation at zero we simply take the limit as t!0+. When = 0 we say that the function is slowly varying. Both theories (t!+1 or t!0+) are equivalents and U(x) is -regularly varying at innite if and only if U(x􀀀1) is (􀀀)- regularly varying at zero. Henceforth, we focus on the theory of regularly variation at innity.
From Theorem 2.2, F() belongs to the max-domain of attraction of the Type II (respectively Type III) extreme value distributions if and only if U(x) = 1 􀀀 F(x) 􀀀 respectively U(x) = 1 􀀀 F 􀀀 xF 􀀀 x .

Branching random walks and extreme of correlated random variables

When investigating the extremal process obtained from random structures of the form (1.12), one must take into account the correlation between variables. In Section 1.2.2, we have dened the overlap function R(; ), which is a natural way to measure the correlation in (1.12). A classical probabilistic model, having similar features is the branching random walks (BRW for short) [AN04]: at the beginning (generation 0), there is a single particle located at zero. This particle gives birth to new particles that are positioned according to a point process L, and dies immediately afterwards. Its children, who form the rst generation, give birth to new particles, that are positioned (with respect to their birth places) according to independent copies of L, and die immediately afterwards; they form the second generation. The system goes on according to the same mechanism. Let BRW(L) denote any BRW dened by i.i.d. copies of L.
Let T be the Galton-Watson tree associated to the genealogical tree of the BRW dened as above. To each point (or individual) of the BRW one can associate a unique vertex w 2 T. Let e 2 T be the root of the Galton-Watson tree, then for a vertex w 2 T, let [[e;w]] denote the shortest path connecting e with w, and jwj the length of this path. We will sometimes write its points [[e;w]] = (e;w1; : : : ;wk) with i = jwij and wk = w. We denote by (w), the position of the individual w 2 T, which is obtained by summing the weights on the edges of [[e;w]], hence, the rightmost position is given by Mn = max jwj=nf(w)g.

Surviving a ballistic absorbing barrier

In contrast with (1.8), in a BRW no constrain concerning the number of particles in the generations is imposed. Yet, one can easily come up with dierent ways to select individuals in a generation and control the population size of the BRW. Selection creates correlation between individuals of same generation and additional dependence in the whole process. In Section 2.2.2, we focus on the so-called M-branching random walk, where only the M-rightmost individuals are chosen to survive. An other classical way to control the population size of a BRW is to add a ballistic absorbing barrier that is, we consider a line moving at a deterministic speed and when a particle lies bellow this moving line it is immediately absorbed (or killed) by it. This model is simply a discrete-time version of the \branching Brownian motion with a barrier » [BBS13, Mai13], for which sharp results have been obtained. A natural question concerning this class of models is whether the system survives or not. Results in this direction were obtained by Gantert, Hu and Shi [GHS11]. They calculate the asymptotic decay for the probability that there exists an innite ray in the BRW that always stays close to (L). We recall that an innite ray [[e;w1]] := fe;w1;w2; : : :g T is an innite collection of vertices (or innite path), such that wi is the parent of wi+1. It represents a family branch in the BRW that has not extinguished, and is parametrized by an element w1 2 @T of the topological boundary @T of the tree.

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The M-branching random walk

Recently, some attention has been paid to models of evolving particle systems under the eect of selection. In this subsection, we will focus on the branching random walks with selection of the M rightmost individuals (the M-BRW) [BG10, BD97, BD99, Mal15]. A continuum version of it (the M branching Brownian motion) has been extensively studied by Maillard [Mai11].
The M-BRW is dened as follows: let L be a point process that has more than one point a.s. (this condition is necessary, since it assures that the process will not extinguish). At generation 0, there are M particles located at zero. The generations are obtained by the following branching/selection step:
Branching: Each particle gives birth to new particles, that are positioned according to independent copies of L.
Selection: One selects the M-rightmost individuals among all children to form the next generation (in average there are MjLj individuals after the branching step).
Hence, the population size is kept constant and equal to M. We denote by M-BRW(L) any M-BRW dened as above. Berard and Gouere [BG10] focused on the binary M-BRW dened by the point process L = p1 + p2 ;

Table of contents :

1 Introduction 
1.1 Stochastic models of front propagation
1.1.1 The F-KPP equation and continuous-time models
1.1.2 Brunet Derrida N-particle system
1.2 First-passage percolation
1.2.1 A directed polymer model.
1.2.2 The correlation structure and the overlap function
1.3 Population models
1.4 Main contribution of the thesis
2 Preliminary Results 
2.1 Extremes of independent random variables
2.1.1 Regularly varying functions and Karamata’s representation
2.1.2 Convergence of moments
2.2 Branching random walks
2.2.1 Surviving a ballistic absorbing barrier
2.2.2 The M-branching random walk
2.3 Ancestral Partition Process and the Coalescent
2.3.1 Weak convergence of ancestral processes
2.3.2 The genealogy of the N-particle system
2.4 Properties of Brunet-Derrida N-particle system
2.4.1 The process seen from the leading edge
2.4.2 The correlation structure
2.5 The Gumbel distribution, a solvable case
2.5.1 Exponential upper tail
2.5.2 Gumbel distribution and last-passage percolation
2.6 Simulations
3 The percolation distribution 
3.1 Introduction
3.2 The two-state percolation distribution
3.2.1 Number of Leading Particles
3.2.2 Front Speed
3.3 Innitly many states distribution
3.3.1 The Distribution of Z( )
3.3.2 Uniform integrability and bounds for T4
3.3.3 Convergence of E [T4] and E 􀀀 X(T4) .
3.3.4 Speed of the N-particle system
3.4 Proof of Theorem 3.3
3.5 Exact diusion constant, percolation distribution
4 Bounded jumps with polynomial density 
4.1 Introduction
4.2 Point processes and branching random walks
4.2.1 Point measures on R
4.2.2 Poisson point processes on ] 􀀀1; 0]
4.2.3 Branching random walks
4.2.4 Elementary properties of Brunet-Derrida’s N-particle system
4.3 Upper bound for the velocity
4.4 Lower bound
4.4.1 Coupling with a branching random walk
4.4.2 Uniform lower bound for the velocities
5 The genealogical aspects of the model 
5.1 Introduction
5.2 Relation with Brunet and Derrida’s model.
5.3 A toy model
5.3.1 Convergence to Kingman’s coalescent E[Y 2 1 ] < 1.
5.3.2 Proof of Theorem 5.1 when 2.
A Appendix A
B Appendix B


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