Instructor: Balazs Rath

Prof. Balint Toth's lecture notes on limit theorems: CLICK

 

Prof. Balint Toth's lecture notes on large deviations (in Hungarian): CLICK

My hand-written, scanned lecture notes:

February 14 (large deviation thm for Binomial r.v.'s, relative entropy, crude Stirling formula): PDF (pages 0-7, )

 

February 14 (exponential Chebyshev's inequality, logarithmic moment generating function, Legendre transform): PDF (pages 7-13)

 

February 21 (large dev. thm. for normal distribution, exponentially tilted distributions, convexity of log.mom.gen function): PDF (pages 14-19, in 2024, we skipped page the proof on 19 (Z is analytic) )

 

February 21 (convolution and tilting, Cramer's theorem, best strategy is to tilt optimally): PDF (pages 19-25)

 

February 22 (heuristics related to Cramer's thm, sum of GEO(p) large deviations, Hoeffding's inequality): PDF (pages 25-32), exercise sheet: PDF

 

February 23 (Bernstein's inequality, Fatou's lemma proof, dominated convergence theorem proof): PDF (pages 32-38, In 2024, we skipped page 33-35 (Bernstein's ineq) )

 

February 23 (c.d.f. properties, weak convergence, max of i.i.d. EXP(1), Gumbel, weak conv. of integer-valued r.v.'s): PDF (pages 39-45, In 2024, we skipped the proof of the claim stated on page 44 )

 

March 6 (BIN and POI, CLT for EXP(1), Stirling's formula, Scheffe's lemma, Slutsky's theorem): PDF (pages 46-52, In 2024, we skipped the proof on page 52 (Slutsky's proof) ), exercise sheet: PDF

 

March 6 (local CLT for BIN(n,1/2) implies global CLT, random walk: reflection principle, limit thm for maximum and hitting time): PDF (pages 53-60)

 

March 13 (Levy distribution is stable, limit thm for return times and local time at zero, arcsine thm for time spent on positive side): PDF (pages 61-72)

 

March 20 (arcsine thm for time spent on positive side and for last visit to origin, tightness, Helly's theorem): PDF (pages 73-80, In 2024, we skipped the proofs on pages 78-80 )

 

March 20 (an equiv. def. of weak convergence, def of characteristic function): PDF (pages 81-86, In 2024, we skipped the proof on page 82-83 )

 

March 20 (First midterm practice): Exercise sheet (solutions in Overleaf)

 

March 27 (properties of char. fn.s, Bochner's thm, derivatives of char. fn. and moments of the r.v., applications: weak law of large numbers ): PDF (pages 87-93, In 2024, we skipped page 88 (Bochner))

 

April 10 (Central limit theorem, probabilistic proof of Weierstrass identity for Gamma function, generating function of r.w. hitting time): PDF (pages 94-100), Exercise sheet and solution of exercise 2(a): PDF

 

April 10 (Fourier inversion formulas, Cauchy distribution): PDF (pages 101-107)

 

April 17 (Lévy's continuity theorem, rooftop function is a char. fn., expressing E(|X|) using the char.fn. of X): PDF (pages 108-114, In 2024, we skipped the claim about E(|X|) on page 113 and its proof )

Audio-lecture: p108, p109, p110, p111, p112, p113, p114.

 

April 17 (statement of Lindeberg's theorem. Applications: number of records, borderline case of CLT): PDF (pages 115-122)

Audio-lecture: p115, p116, p117, p118, p119, p120, p121, p122.

 

April 24 (proof of Lindeberg's thm, Lindeberg is not applicable to coupon collector problem, Euler product formula for Riemann zeta function): PDF (pages 123-128, In 2024, we skipped the Euler product formula (page 128))

Audio-lecture: p123, p124, p125, p126, p127, p128.

 

April 24 (Erdos-Kac theorem (i.e., CLT for the number of prime divisors), method of moments): PDF (pages 129-139, In 2024, we skipped the proofs on page 134 and on 137-139 )

Audio-lecture: p129, p130, p131, p132, p133, p134, p135, p136, p137, p138, p139.

 

May 8 (Stable distributions and their relation to weak limits of sums of scaled shifted i.i.d. r.v.'s, characterization thm of symmetric stable laws): PDF (pages 140-147, In 2024, we skipped the proof on page 143-145)

 

May 8 (Second midterm practice): Exercise sheet and the corresponding solutions: CLICK

 

May 15 (proof of characterization theorem of symmetric stable laws): PDF (pages 148-155, In 2024, we skipped the proof on page 151-155)

 

May 15 / May 22 (Holtsmark's problem): PDF (pages 156-163)

 

May 15 / May 22 (comparing weak limits of sums and maxima, harmonic mean weakly converges to Cauchy, non-symmetric stable laws and limit thms ): PDF (pages 164-173, In 2024, we skipped non-symmetric stable laws (page 169-173))