Limit theorems and large deviation theorems of probability
2024 Spring
- Instructor: Balazs Rath
- Prof. Balint Toth's lecture notes on limit theorems:
CLICK
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- 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, )
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- February 14 (exponential Chebyshev's inequality, logarithmic
moment generating function, Legendre transform):
PDF (pages 7-13)
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- 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) )
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- February 21 (convolution and tilting, Cramer's theorem, best
strategy is to tilt optimally):
PDF (pages 19-25)
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- February 22 (heuristics related to Cramer's thm, sum of GEO(p)
large deviations, Hoeffding's inequality):
PDF (pages 25-32), exercise sheet: PDF
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- 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) )
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- 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
)
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- 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
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- 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)
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- 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)
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- 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 )
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- 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 )
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- March 20 (First midterm practice):
Exercise sheet (solutions in Overleaf)
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- 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))
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- 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
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- April 10 (Fourier inversion formulas, Cauchy distribution):
PDF (pages 101-107)
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- 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.
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- 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.
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- 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.
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- 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.
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- 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)
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- May 8 (Second midterm practice):
Exercise sheet and the corresponding solutions: CLICK
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- May 15 (proof of characterization theorem of symmetric stable laws):
PDF (pages 148-155, In 2024, we skipped
the proof on page 151-155)
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- May 15 / May 22 (Holtsmark's problem):
PDF (pages 156-163)
- Audio-lecture:
p156,
p157,
p158,
p159,
p160,
p161,
p162,
p163.
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- 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))
- Audio-lecture:
p164,
p165,
p166,
p167,
p168,
p169,
p170,
p171,
p172,
p173.
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