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

Convergence of Probability Measures

Fractional Differential Equations

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

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

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).

The main purpose of this paper is to study the problem of determining initial condition of nonlinear parabolic equation from noisy observations of the final condition. We introduce a regularized method to establish an approximate solution. We prove an upper bound on the rate of convergence of the mean integrated squared error.

Convergence rates in expectation for a nonlinear backward parabolic equation with Gaussian white noise

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data u(T,x)=φ(x)$$ u\left(T,x\right)&amp;amp;#x0003D;\varphi (x) $$ and the spectrum of the fractional Laplacian operator (−Δ)s/2$$ {\left(-\Delta \right)}&amp;amp;#x0005E;{s/2} $$ (in a bounded domain 𝕏⊂ℝd , 0

On a terminal value problem for stochastic space‐time fractional wave equations

This paper studies a time-changed stochastic control problem, where the underlying stochastic process is a Levy noise time-changed by an inverse subordinator. We establish a maximum principle theory for the time-changed stochastic control problem. We also prove the existence and uniqueness of the corresponding time-changed backward stochastic differential equation involved in the stochastic control problem. Some examples are provided for illustration.

Time-changed Stochastic Control Problem and its Maximum Principle Theory.

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, ¶)$ be a sequence of random variables For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$ It is proved that, if there exist a nondecreasing function $\varphi: \R_+\to \R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge 2$ such that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}} {\varphi(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty} \frac{S_{0, n}} {\varphi(n)} = 0\qquad \hbox{as} $$ This extends Theorem 1 in Levental, Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes

Erkan Nane

Papers

Convergence rates in expectation for a nonlinear backward parabolic equation with Gaussian white noise

On a terminal value problem for stochastic space‐time fractional wave equations

Time-changed Stochastic Control Problem and its Maximum Principle Theory.

A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes

Seismo-ionospheric anomalies related to the Mw 6.6, July 20, 2017, earthquake in Bodrum, Turkey