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

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

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

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

Preface 1 Introduction 1.1 The traditional diffusion model 1.2 Fractional diffusion 2 Fractional Derivatives 2.1 The Grunwald formula 2.2 More fractional derivatives 2.3 The Caputo derivative 2.4 Time-fractional diffusion 3 Stable Limit Distributions 3.1 Infinitely divisible laws 3.2 Stable characteristic functions 3.3 Semigroups 3.4 Poisson approximation 3.5 Shifted Poisson approximation 3.6 Triangular arrays 3.7 One-sided stable limits 3.8 Two-sided stable limits 4 Continuous Time Random Walks 4.1 Regular variation 4.2 Stable Central Limit Theorem 4.3 Continuous time random walks 4.4 Convergence in Skorokhod space 4.5 CTRW governing equations 5 Computations in R 5.1 R codes for fractional diffusion 5.2 Sample path simulations 6 Vector Fractional Diffusion 6.1 Vector random walks 6.2 Vector random walks with heavy tails 6.3 Triangular arrays of random vectors 6.4 Stable random vectors 6.5 Vector fractional diffusion equation 6.6 Operator stable laws 6.7 Operator regular variation 6.8 Generalized domains of attraction 7 Applications and Extensions 7.1 LePage Series Representation 7.2 Tempered stable laws 7.3 Tempered fractional derivatives 7.4 Pearson Diffusion 7.5 Classification of Pearson diffusions 7.6 Spectral representations of the solutions of Kolmogorov equations 7.7 Fractional Brownian motion 7.8 Fractional random fields 7.9 Applications of fractional diffusion 7.10 Applications of vector fractional diffusion Bibliography Index

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Stochastic Models for Fractional Calculus

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space-time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.

http://prac.im.pwr.edu.pl/~hugo/publ/AJurlewiczPKernMMeerschaertHPSchefler_camwa2011.pdf

Fractional governing equations for coupled random walks

Let X={X(t)}t≥0 be an operator semistable Levy process in ℝd with exponent E, where E is an invertible linear operator on ℝd and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao (Stoch. Process. Appl. 115, 55–75, 2005) for the special case of an operator stable (selfsimilar) Levy process, for an arbitrary Borel set B⊆ℝ+ we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.

The Hausdorff Dimension of Operator Semistable Lévy Processes

We study sample path deviations of the Wiener process from three different representations of its bridge: anticipative version, integral representation and space–time transform. Although these representations of the Wiener bridge are equal in law, their sample path behavior is quite different. Our results nicely demonstrate this fact. We calculate and compare the expected absolute, quadratic and conditional quadratic path deviations of the different representations of the Wiener bridge from the original Wiener process. It is further shown that the presented qualitative behavior of sample path deviations is not restricted only to the Wiener process and its bridges. Sample path deviations of the Ornstein–Uhlenbeck process from its bridge versions are also considered and we give some quantitative answers also in this case.

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Sample path deviations of the Wiener and the Ornstein–Uhlenbeck process from its bridges

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property for general multidimensional linear process bridges. We specialize our results for the one-dimensional case; especially, we study one-dimensional OrnsteinUhlenbeck bridges.

http://reh.math.uni-duesseldorf.de/~stoch/Kern/bridgerep.pdf

Representations of multidimensional linear process bridges

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property for general multidimensional linear process bridges. We specialize our results for the one-dimensional case; especially, we study one-dimensional Ornstein-Uhlenbeck bridges.

Peter Kern

Papers

Fractional governing equations for coupled random walks

The Hausdorff Dimension of Operator Semistable Lévy Processes

Sample path deviations of the Wiener and the Ornstein–Uhlenbeck process from its bridges

Representations of multidimensional linear process bridges

Representations of multidimensional linear process bridges