Instructor: Balazs Rath

Rick Durrett: Essentials of Stochastic Processes: CLICK

Simon Karoly Sztochasztikus Folyamatok jegyzete (magyar nyelven): KLIKK

English-Hungarian Math Dictionary (General): PDF

My hand-written, scanned lecture notes:

Sept 4 (gambler's ruin, volleyball, matrix powers and eigenvalues/eigenvectors): PDF (pages 1-8); Dictionary: PDF

 

Sept 8 (stoch. proc. def, Markov chain def, GEO expectation, simple random walk on graph): PDF (pages 9-17); Dictionary: PDF

 

Sept 11 (multistep transition prob., Chapman-Kolmogorov eq., non-bactracking r.w., gambler's ruin): PDF (pages 16-23); Dictionary: PDF

 

Sept 15 (stationary distribution, uniqueness, law of large numbers for Markov chains, examples): PDF (pages 24-31); Dictionary: PDF

 

Sept 18 (two-step chain, detailed balance, birth/death chains, Ehrenfest chain, time-reversal): PDF (pages 32-40); Dictionary: PDF

 

Sept 22 (reversible M.c., r.w. on graph, doubly stoch. matrix, hitting prob. for birth/death): PDF (pages 38-46); Dictionary: PDF

 

Sept 25 (exit times for 1-dim simple r.w., lin. alg. proof of existence/uniqueness of stationary distr., conv. to stationarity): PDF (pages 47-54); Dictionary: PDF

 

Sept 29 (periodicity of Markov chains, aperiodicity implies convergence to stationarity): PDF (pages 55-59); Dictionary: PDF

 

Oct 2 (discussion of Markov property, classification of states of finite state space M.c.): PDF (pages 60-66); Dictionary: PDF

 

Oct 6 (recurrent/transient states in finite state space, stopping time, strong Markov, renewal theory): PDF (pages 67-74); Dictionary: PDF

 

Oct 9 (M.c. with countably infinite state space, recurrence/transience, birth/death chains): PDF (pages 75-82); Dictionary: PDF

 

Oct 13 (expected number of visits before return gives a stationary measure, def of positive recurrence, asymptotic frequency of returns): PDF (pages 83-91); Dictionary: PDF

 

Oct 16 (pos. recurrence, law of large numbers for M.c., Galton-Watson branching process, generating function tricks): PDF (pages 92-99); Dictionary: PDF

 

Oct 20 (extinction probability of branching proc., Poisson distribution: merging, splitting/coloring): PDF (pages 100-109); Dictionary: PDF

 

Oct 27 (Geiger counter, def of Poisson point process (PPP), Gamma distribution): PDF (pages 110-114); Dictionary: PDF

 

Oct 30 (construction of inhomogeneous PPP, examples): PDF (pages 115-121); Dictionary: PDF

 

Nov 3 (examples, coloring and superposition of PPP, exponential races, compound PPP): PDF (pages 122-129); Dictionary: PDF

 

Nov 6 (coloring/merging examples, sum of GEO number of EXP, continuous-time MC def, infinitesimal generator): PDF (pages 130-137); Dictionary: PDF

 

Nov 10 (Kolmogorov fw/bw eq, matrix exponential, stationary distribution, construction using i.i.d. EXP holding times): PDF (pages 138-145); Dictionary: PDF

 

Nov 13 (convergence to stationarity, jump times and jump locations, competing clocks, PPP as a MC): PDF (pages 146-154); Dictionary: PDF

 

Nov 17 (detailed balance, birth-death proc, M/M/s queue, Burke's thm, embedded M.c., time spent at M/M/1 queue): PDF (pages 155-163); Dictionary: PDF

 

Nov 20 (examples, martingale def, conditional expectation def, SRW is a martingale): PDF (pages 164-171); Dictionary: PDF

 

Nov 27 (tower property, martingale w.r.t. the filtration of another process, harmonic function of M.c. is a martingale): PDF (pages 172-178); Dictionary: PDF

 

Dec 1 (Wright-Fisher martingale, biased random walk exponential martingale, branching process martingale): PDF (pages 179-182); Dictionary: PDF

 

Dec 4 (submartingale, supermartingale, separation of known info, predictable trading strategy, optional stopping thm): PDF (pages 182-189); Dictionary: PDF

 

Dec 8 (applications of optional stopping thm: random walk hitting probabilities, hitting times, Wright-Fisher absorbtion time): PDF (pages 190-196); Dictionary: PDF