AIS Stochastic Processes (2018)

School Convener(s)

• Discrete Markov chains with countable state space, Examples including 2-state chain, birth and death chain, renewal chain, Ehrenfest chain, card shufing, etc.Classifcaaon of states, recurrence and transience; absorbing states, irreducibility, decomposiaon of state space into irreducible classes, Examples.
• Absorbing chains, absorpaon probabiliaes and mean absorpaon ame, fundamental matrix Staaonary distribuaons, limit theorems, posiave and null recurrence, reversible chains. Periodicity, limit theorems for aperiodic irreducible chains. Introducaon to MCMC.

Antar Bandyapadhyay

• Measurable funcaons, basic properaes, sigma-felds generated by funcaons, integraaon of measurable funcaons, properaes of integrals, MCT, Fatou’s Lemma, DCT. Chebyshev’s, Holder’s and Minkowski’s inequaliaes. LLp spaces. Finite product of measurable spaces, construcaon of product measures, Fubinis theorem. Probability spaces, random variables and random vectors, expected value and its properaes.

• Review of basic probability Theory, Random walks

• Independence. Various modes of convergence and their relaaon. The Borel-Cantelli lemmas. Weak Law of large numbers for i.i.d. fnite mean case. Kolmogorov 0-1 law. Strong law of large numbers for i.i.d. case. Characterisac funcaons and its basic properaes, inversion formula, Levy’s conanuity theorem. Lindeberg CLT, CLT for i.i.d. fnite variance case, Lyapunov CLT.

• Metric spaces, open/closed sets, sequences, compactness, completeness, conanuous funcaons and homeomorphisms, connectedness, product spaces, completeness of C[0, 1] and L p spaces

• Doing integraaon beyond Riemann theory, infnite tosses of a fair coin Fields, sigma-felds, measures, sigma-fnite/fnite/probability measures, properaes, statement of Caratheodory extension theorem (outline of idea, if ame permits). Monotone class theorem, Dynkin’s pi-lambda theorem. Radon measures on fnite-dimensional Borel sigma-feld, distribuaon funcaons,
  correspondence between probability measures on Borel sigma-feld and probability distribuaon funcaons.

• Analyac funcaons, Cauchy-Riemann equaaons, polynomials, exponenaal and trigonometric funcaons Contour integraaon, Power series representaaon of analyac funcaons, Liouville’s theorem, Cauchy integral formula, Cauchy’s theorem, Morera’s theorem, Cauchy-Goursat theorem Singulariaes, Laurent Series expansion, Cauchy residue formula, residue calculus. Meromorphic funcaons, Rouche’s theorem.

• Random walk: Recurrence and transience, Generaang funcaon, convoluaon, fxed point of generaaon funcaon, convergence of p.g.f., Branching processes, Gambler’s Ruin problem, Arc Sine law