Many subtle definitions, such as interior point, accumulation point, open set, closed set, are all based on the concepts of distance and neighborhoods.  The beauty is mathematics is you could always go further with good understanding of the fundamental ideas by going into general. The metric space is a good example of the generalization of ideas, from distance to metric . Further to the metric space. Most concepts which have been defined under the framework of distance and neighborhood thus can be extended to this more general setting. This is one way mathematics extended to more general setting and developed into new math branch.  

This Analysis book by Steven is a great introduction book covers most concrete analysis areas. By concrete, I mean the contents (Analysis) is not abstract, but fundamental for going deeper to higher analysis. It is also an well organized book with lots of example which clarifies the underlying definition or ideas which build up the analysis. It is a good introduction book for peers who are just get in touch with Analysis or even for graduate who are not confident enough with their prior knowledge on Analysis, it even helpful to refresh what is Analysis really about and what is use of it. I personally like the book very much. 

DEFINITION 1 A probability space (\Omega, \mathcal F, P) is a triplet of a set \Omega, a family \mathcal F of subsets \Omega, and a mapping P from \mathcal F into \mathbb R satisfying the following conditions: Read the rest of this entry »

Convergence in distribution

Suppose that F1F2, … is a sequence of cumulative distribution functions corresponding to random variables X1X2, …, and that F is a distribution function corresponding to a random variable X. We say that the sequence Xn converges towards X in distribution, if

\lim_{n\rightarrow\infty} F_n(a) = F(a),

for every real number a at which F is continuous. Since F(a) = Pr(X ≤ a), this means that the probability that the value of X is in a given range is very similar to the probability that the value of Xn is in that range, provided n is sufficiently large. Convergence in distribution is often denoted by adding the letter \mathcal D over an arrow indicating convergence:

X_n \, \xrightarrow{\mathcal D} \, X

Small d is also possible, although less common.

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In mathematical analysisdistributions (also known as generalized functions) are objects which generalize functions and probability distributions. They extend the concept of derivative to all integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution.

“Generalized functions” were introduced by Sergei Sobolev in 1935. They were independently introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.

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This book brings reader the fully-fledged type of thinking used by professional mathematicians. It develops the more formal approach as a natural outgrowth of the pattern of underlying ideas, building on a school-mathematics background to develop the viewpoint of an advanced practicing mathematician. It covers the nature of mathematical thinking; a review of the intuitive development of familiar number systems; sets, relations, functions; an introduction to logic as used by practicing mathematicians, methods of proof (including how mathematical proof is written); development of axiomatic number systems from natural numbers and proof by induction to the construction of the real and complex numbers; the real number as a complete ordered field; cardinal numbers; foundations in retrospect.It is indeed an interesting and helpful book for readers in transition from ‘school mathematics’ to be a professional mathematician. 

The following articles are about Rough path theory which is developed by T. Lyons from Oxford University:

  1.  Lyons-Victoir extension
  2. An introduction to Rough paths

For more reference on Rough paths, one could visit Rough paths.

The following is collections of articles published on subject of Tanaka’s formula:

  1.  Tanaka’s formula for symmetric levy processe

The following articles are a collection of local time related publications:

  1. Continuity of local time for Levy processes 
  2. Continuity of local time for markov processes
  3. Joint continuity of local time for markov process
  4. Large deviation of local time for levy processes
  5. Limit theorem and variation properties for fractional derivatives of the local time of a stable process
  6. Local time of markov processes approximated by a generalized iterated brownian motion
  7. Local time for Markov processes
  8. local time and related properties of multidimensional iterated brownian motion
  9. Occupation densities
  10. On the barlow yor inequality of local time
  11. Semi-martingale inequality and local time
  12. Unbounded local time
  13. Two parameter p,q variation path and integration of local time
  14. Generalized Itǒ Formulae and Space-Time Lebesgue–Stieltjes Integrals of Local Times

The following article I wrote is based on Bertoin’s Levy Processes. It reviews on how potential theory has been used to construct Occupation densities . It is well known that there are different constructions and definitions of local times corresponding to different classes of stochastic processes. For a large panorama of such definitions, refer to Occupation densities