다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Convergence Of Probability Measures

Invariance principles for Galton-Watson trees conditioned on the number of leaves

We introduce a class of random compact metric spaces $\mathscr{L}_{\alpha}$ indexed by  $\alpha~\in(1,2)$ and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be viewed as dual graphs of  $\alpha$-stable Levy trees. We study their properties and prove in particular that the Hausdorff dimension of $ \mathscr{L}_{\alpha}$ is almost surely equal to $\alpha$. We also show that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter $ \frac{3}{2}$ is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.

/pdf/random-stable-looptrees-3izrzqchhs.pdf

Random stable looptrees

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton–Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index θ ∈ (1, 2], conditioned on having total progeny n, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive Levy process of index θ. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly n and the conditional probability of having total progeny at least n. This new method is robust and can be adapted to establish invariance theorems for Galton–Watson trees having n vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.

A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Racz (9) concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric struc- ture of nodes of large degrees in a plane version of Barabasi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdor sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdor dimension 2. Resume (Limites d'echelle et ontogenese des arbres construits par attachement preferentiel) Nous nous interessons au comportement asymptotique d'arbres aleatoires construits par attachement preferentiel lineaire, qui sont aussi connus dans la litterature sous le nom d'arbres de Barabasi-Albert ou encore arbres plans recursifs. Nous validons une conjecture de Bubeck, Mossel & Racz relative a l'influence de l'arbre initial sur le comportement asymptotique de ces arbres. Separement, nous etudions la structure geometrique des sommets de grand degre dans la version planaire des arbres de Barabasi-Albert en considerant leurs « arbres a boucles ». Lorsque le nombre de sommets croit, nous prouvons que ces arbres a boucles, convenablement mis a l'echelle, convergent au sens de Gromov-Hausdor vers un espace metrique compact aleatoire, que nous appelons « l'arbre a boucles brownien ». Ce dernier est construit comme un espace quotient de l'arbre continu brownien d'Aldous, et nous prouvons que sa dimension de Hausdor vaut 2 presque surement.

/pdf/scaling-limits-and-influence-of-the-seed-graph-in-5aybpksaz3.pdf

Scaling limits and influence of the seed graph in preferential attachment trees

We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution towards Aldous' Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton-Watson tree structure. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 236-260, 2014

Igor Kortchemski

Papers

Invariance principles for Galton-Watson trees conditioned on the number of leaves

Random stable looptrees

A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees

Scaling limits and influence of the seed graph in preferential attachment trees

Random non-crossing plane configurations: A conditioned Galton-Watson tree approach