Genealogies of regular exchangeable coalescents with applications to sampling

Vlada Limic

Displaying similar documents to “Genealogies of regular exchangeable coalescents with applications to sampling”

Asymptotic sampling formulae for $𝛬$ -coalescents

Julien Berestycki, Nathanaël Berestycki, Vlada Limic (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $𝛬$ -coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty$ . Some of our results hold in the case of...

On the distance between ⟨X⟩ and $L^{\infty}$ in the space of continuous BMO-martingales

Litan Yan, Norihiko Kazamaki (2005)

Studia Mathematica

Similarity:

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${| | X | |}_{B M O} \equiv s u p_{T} | | E [| X_{\infty} - X_{T} | | ℱ_{T} {] | |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b (X) = b > 0 : s u p_{T} | | E [e x p (b ² (⟨ X ⟩_{\infty} - ⟨ X ⟩_{T})) | ℱ_{T}] {| |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q (X) ₜ = E [⟨ X ⟩_{\infty} | ℱ ₜ] - E [⟨ X ⟩_{\infty} | ℱ ₀]$ . We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty}$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1 / 4 d ₁ (q (X), L^{\infty}) \leq b (X) \leq 4 / d ₁ (q (X), L^{\infty})$ hold for every continuous BMO-martingale X. ...

Polar wavelets and associated Littlewood-Paley theory

Epperson Jay, Frazier Michael

Similarity:

Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = \sum_{μ, k, m} ⟨ f, φ_{μ k m} ⟩ ψ_{μ k m}$ , in which each function $φ_{μ k m}$ and $ψ_{μ k m}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

Noncommutative fractional integrals

Narcisse Randrianantoanina, Lian Wu (2015)

Studia Mathematica

Similarity:

Let ℳ be a hyperfinite finite von Nemann algebra and ${(ℳ_{k})}_{k \geq 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${(ℳ_{k})}_{k \geq 1}$ . For a finite noncommutative martingale $x = {(x_{k})}_{1 \leq k \leq n} \subseteq L ₁ (ℳ)$ adapted to ${(ℳ_{k})}_{k \geq 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{α} x = \sum_{k = 1}^{n} ζ_{k}^{α} d x_{k}$ for an appropriate sequence ${(ζ_{k})}_{k \geq 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

Pointwise multipliers on martingale Campanato spaces

Eiichi Nakai, Gaku Sadasue (2014)

Studia Mathematica

Similarity:

We introduce generalized Campanato spaces $_{p, ϕ}$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then $_{p, ϕ} = B M O$ . We give a characterization of the set of all pointwise multipliers on $_{p, ϕ}$ .

On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi (2013)

Colloquium Mathematicae

Similarity:

Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f = {(f ₙ)}_{n \in ℤ ₊} \in ℳ$ , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${| | M g | |}_{X} {\leq | | M f | |}_{X}$ , then ${| | S g | |}_{X} {\leq C | | S f | |}_{X}$ , where C is a constant independent of f and g.

A Note on the Burkholder-Rosenthal Inequality

Adam Osękowski (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $∥ \sum_{k = 0}^{\infty} d f_{k} ∥_{p} \leq C_{p} ∥ (\sum_{k = 0}^{\infty} (| d f_{k} | ² | ℱ_{k - 1} {))}^{1 / 2} ∥_{p} + ∥ (\sum_{k = 0}^{\infty} | d f_{k} {|^{p})}^{1 / p} ∥_{p},$ with $C_{p} = O (p / l n p)$ as p → ∞.

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

Similarity:

Let ${X_{n, j}, 1 \leq j \leq m (n), n \geq 1}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ of the dependent bootstrap samples ${{X_{n, j}^{*}, 1 \leq j \leq m (n)}, n \geq 1}$ arising from a sequence of i.i.d. random variables ${X_{n}, n \geq 1}$ .

Moment Inequality for the Martingale Square Function

Adam Osękowski (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Consider the sequence ${(C ₙ)}_{n \geq 1}$ of positive numbers defined by C₁ = 1 and $C_{n + 1} = 1 + C ₙ ² / 4$ , n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Branching Processes in Random Environment (BPREs) $(Z_{n} : n \geq 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $ℙ (1 \leq Z_{n} \leq k | Z_{0} = i)$ , $k, i \in ℕ$ as $n \to \infty$ . More precisely, we characterize...

Limit distributions for multitype branching processes of $m$ -ary search trees

Brigitte Chauvin, Quansheng Liu, Nicolas Pouyanne (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Let $m \geq 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m \leq 26$ , the asymptotic behavior of the process is Gaussian, but for $m \geq 27$ it is no longer Gaussian and a limit $W^{D T}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Large scale behaviour of the spatial $𝛬$ -Fleming–Viot process

N. Berestycki, A. M. Etheridge, A. Véber (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We consider the spatial $𝛬$ -Fleming–Viot process model ( (2010) 162–216) for frequencies of genetic types in a population living in $ℝ^{d}$ , in the special case in which there are just two types of individuals, labelled $0$ and $1$ . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$ . We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the...

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

This paper is concerned with the small time behaviour of a Lévy process $X$ . In particular, we investigate theof the times, ${\bar{T}}_{b} (r)$ and $T_{b}^{*} (r)$ , at which $X$ , started with $X_{0} = 0$ , first leaves the space-time regions ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (one-sided exit), or ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (two-sided exit), $0 \leq b l t; 1$ , as $r ↓ 0$ . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^{p}$ . In many instances these are...

Random walks in ${(ℤ_{+})}^{2}$ with non-zero drift absorbed at the axes

Irina Kurkova, Kilian Raschel (2011)

Bulletin de la Société Mathématique de France

Similarity:

Spatially homogeneous random walks in ${(ℤ_{+})}^{2}$ with non-zero jump probabilities at distance at most $1$ , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes. ...

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let Ω be a countable infinite product $Ω ₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda, G. Morando (2011)

Bulletin de la Société Mathématique de France

Similarity:

In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$ . Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$ . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$ . Second,...

A lower bound in the law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2014)

Studia Mathematica

Similarity:

We prove a lower bound in a law of the iterated logarithm for sums of the form $\sum_{k = 1}^{N} a_{k} f (n_{k} x + c_{k})$ where f satisfies certain conditions and the $n_{k}$ satisfy the Hadamard gap condition $n_{k + 1} / n_{k} \geq q > 1$ .

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

In the present work, we consider spectrally positive Lévy processes $(X_{t}, t \geq 0)$ not drifting to $+ \infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$ ) before hitting $0$ . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $ℙ_{x}^{☆}$ of this conditioned process (starting at $x g t; 0$ ) is defined as a Doob $h$ -transform via a martingale. For Lévy processes with infinite variation paths,...

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

Similarity:

We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

Similarity:

We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.

The Gibbs phenomenon and Lebesgue constants for regular $A (λ)$ means

V. Swaminathan (1977)

Matematički Vesnik

Similarity: