Writing Principles and the Direction of the Book

Directed by Haobai Wang, this book begins with historical mathematical problems and the limits of existing tools, explaining why new concepts become necessary. Definitions, derivations, theorems, proofs and counterexamples serve a continuous development of ideas.

The book takes the breadth of modern probability as a reference, pursues depth of mathematical mechanism, and requires correct statements, hypotheses and conclusions. Narrative continuity is indispensable. Lengthy technical work may enter proof topics, but a proof idea must not be presented as a complete proof.

Chapters 1 and 2 provide the established narrative benchmark. Brownian motion remains Chapter 4: random walks and finite-dimensional laws raise the question of an entire path, followed by measure, integration, convergence and infinite experiments.

The later route connects conditional expectation, probability kernels, independent sequences, modern limit theorems, weak convergence, martingales and processes, then stochastic integration, Lévy theory, exchangeability, random measures and random geometry. Problems and proofs determine chapter capacity; a fixed chapter count must not compress the content.

Historical facts and modern reconstructions are distinguished, and sources remain attached to references and third-party materials. Author adoption, mathematical review, historical verification and engineering builds are independent states; a preview does not automatically become a formal publication or academically finalized text.

Only reader-facing writing principles appear here. Private writing records, prompts, unpublished research and internal discussions are not included in public webpages or downloads.