## Further Topics in Probability, Spring 2015

 Márton Balázs Email: m.balazs@our_city.ac.countrycode Tel: +44 (0)117 928-7991 Office: Maths 3.7 Office hours: Mondays 14:30 - 15:30 on teaching weeks, Maths 3.7 except for the 16th of February, which is cancelled.

Below is a detailed schedule. Topics of future events are plans, and can change. Topics of past events serve as log.
Homeworks are/will also be posted here: just click those with a link below. They are due in class, or in the blue locker with my name on the ground floor of the Main Maths Building.
Homework solutions will appear on Blackboard, please email me if you have problem accessing these.

I was thinking about going over the ingenious proof of the SLLN by N. Etemadi, but decided to stay with the classical way instead. You are welcome to check out his way of doing it!

Came across this illustration of Jensen's inequality: square-root of the average ≠ average of the square-root.

Day Topics Homework due:
Mon 26 Jan, 13:00, Maths SM4 Review of the unit, basic discrete distributions --
Tue 27 Jan, 9:00, Maths SM4 Convolution (discrete cases) --
Tue 27 Jan, 14:00, Phys 3.34 Ex. Class (Neg.Binom and Uniform convolution; Residue Thm.) --
Thu 29 Jan, 15:00, Med C44 Convolution (Gaussian) --
Mon 2 Feb, 13:00, Maths SM4 Convolution (Cauchy) --
Tue 3 Feb 9:00, Maths SM4 Convolution (Cauchy, Exponential) --
Tue 3 Feb, 14:00, Phys 3.34 Ex. Class (Convolutions) HW1
Thu 5 Feb, 15:00, Med C44 Poisson process, Chi square; Generating functions (properties) --
Mon 9 Feb, 13:00, Maths SM4 Generating functions (examples; of convolutions) --
Tue 10 Feb, 9:00, Maths SM4 Generating functions (random no. of summands, Galton-Watson process) --
Tue 10 Feb, 14:00, Phys 3.34 Ex. Class (Generating functions) --
Thu 12 Feb, 15:00, Med C44 Generating functions (Galton-Watson process) --
Mon 16 Feb, 13:00, Maths SM4 Generating functions (random walk: level 1 hitting time) --
Tue 17 Feb, 9:00, Maths SM4 Generating functions (random walk, weak convergence) --
Tue 17 Feb, 14:00, Phys 3.34 Ex. Class (Generating functions) HW2
Thu 19 Feb, 15:00, Med C44 Generating functions (Poisson approximation) --
Mon 23 Feb, 13:00, Maths SM4 Weak Law of Large Numbers, Stirling's formula --
Tue 24 Feb, 9:00, Maths SM4 Normal distribution, DeMoivre Laplace CLT --
Tue 24 Feb, 14:00, Phys 3.34 Ex. Class (Generating fcts, WLLN, Stirling's formula) --
Thu 26 Feb, 15:00, Med C44 DeMoivre Laplace CLT --
Mon 2 Mar, 13:00, Maths SM4 Measure Theory (basic notions, probability) --
Tue 3 Mar, 9:00, Maths SM4 Measure Theory --
Tue 3 Mar, 14:00, Phys 3.34 Ex. Class (DeMoivre Laplace CLT), Weierstrass Approx. Thm. HW3
Thu 5 Mar, 15:00, Med C44 Probabilistic tools --
Mon 9 Mar, 13:00, Maths SM4 Probabilistic tools --
Tue 10 Mar, 9:00, Maths SM4 Notions of convergence --
Tue 10 Mar, 14:00, Phys 3.34 Ex. Class (Types of convergences) --
Thu 12 Mar, 15:00, Med C44 Convergences; Strong Law of Large Numbers (Kolmogorov's ineq.) --
Mon 16 Mar, 13:00, Maths SM4 Strong Law of Large Numbers (Kolmogorov-Khinchin, Toeplitz, Kronecker) --
Tue 17 Mar, 9:00, Maths SM4 Kolmogorov's Thm, SLLN --
Tue 17 Mar, 14:00, Phys 3.34 Ex. Class (Convergences, SLLN) HW4
Thu 19 Mar, 15:00, Med C44 Characteristic Functions --
Mon 23 Mar, 13:00, Maths SM4 Characteristic Functions (Inversion formula) --
Tue 24 Mar, 9:00, Maths SM4 Inversion formula, Weak Convergence --
Tue 24 Mar, 14:00, Phys 3.34 Ex. Class (Characteristic functions, Weak convergence) --
Thu 26 Mar, 15:00, Med C44 Weak Convergence, CLT (Prokhorov's Thm) --
Easter vacation
Mon 20 Apr, 13:00, Maths SM4 Weak Convergence, CLT (Continuity Lemma) --
Tue 21 Apr, 9:00, Maths SM4 Weak Convergence, WLLN, CLT --
Tue 21 Apr, 14:00, Phys 3.34 Ex. Class (LLN, Characteristic functions, Lindeberg Thm) HW5
Thu 23 Apr, 15:00, Med C44 Weak Convergence, CLT (Comments on the CLT) --
Mon 27 Apr, 13:00, Maths SM4 Renewal Theory (LLN, Delayed renewals) --
Tue 28 Apr, 9:00, Maths SM4 Renewal Theory (Renewal theorems, stationary renewals) --
Tue 28 Apr, 14:00, Phys 3.34 Ex. Class (Age, renewal equation, renewal theorems) --
Thu 30 Apr, 15:00, Maths C44 Renewal Theory (residual time, proofs) --
Mon 4 May Bank holiday
Tue 5 May, 9:00, Maths SM4 Renewal Theory (Proofs) --
Tue 5 May, 14:00, Phys 3.34 Ex. Class (Gaps, terminating renewal) --
Thu 7 May, 15:00, Med C44 Renewal Theory (Proofs, renewal paradox, length-biased sampling) --
Fri 8 May, 16:00: HW6 deadline only HW6
Tue 12 May, 10:00, Maths SM3 Q&A session --