## Further Topics in Probability, Spring 2018

 Márton Balázs Email: m.balazs@our_city.ac.countrycode Tel: +44 (0)117 928-7991 Office: Maths 3.7 Drop in Sessions: Tuesdays 13:00 - 14:00 on teaching weeks, Maths 3.7 Q&A Session: Friday 25th May, 14:00, SM4. Please prepare with questions.

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:
Tue 23 Jan Review of the unit, basic discrete distributions --
Thu 25 Jan Basic discrete distributions; convolution (discrete cases) --
Fri 26 Jan Ex. class (Neg.Bin convolution, continuous r.v., Normal distribution) --
Tue 30 Jan Convolution (Gaussian) --
Thu 1 Feb Convolution (Cauchy) --
Fri 2 Feb Convolution (Cauchy) HW1 (sol. on Bb.)
Tue 6 Feb Gamma, Chi square distributions --
Thu 8 Feb Poisson process. Generating functions (properties) --
Fri 9 Feb Ex. class (convolutions) --
Tue 13 Feb Generating functions (random no. of summands) --
Thu 15 Feb Generating functions (Galton-Watson process) --
Fri 16 Feb Generating functions (Critical G-W process; random walk) HW2 (sol. on Bb.)
Tue 20 Feb Generating functions (random walk: level 1 hitting time) --
Thu 22 Feb Generating functions (weak convergence) --
Fri 23 Feb Ex. class (fun with generating functions) --
Tue 27 Feb Weak convergence examples, Weak Law of Large Numbers --
Thu 1 Mar Weak Law of Large Numbers, Stirling's formula -- Cancelled --
Fri 2 Mar Stirling's formula, DeMoivre Laplace CLT -- Cancelled HW3 -- Due: 5th (sol. on Bb.)
Tue 6 Mar Weak Law of Large Numbers, Stirling's formula --
Thu 8 Mar Weierstrass' Approximation Thm, DeMoivre Laplace CLT --
Thu 8 Mar DeMoivre Laplace CLT --
Fri 9 Mar Ex. class (Generating functions: RW probabilities) --
Tue 13 Mar Measure Theory (basic notions, probability) --
Tue 13 Mar Measure Theory --
Thu 15 Mar Probabilistic tools --
Fri 16 Mar Probabilistic inequalities HW4 (sol. on Bb.)
Tue 20 Mar Convergences --
Thu 22 Mar Strong Law of Large Numbers (Kolmogorov's ineq., Kolmogorov-Khinchin) --
Fri 23 Mar Ex. class (convergence types) --
Easter vacation
Tue 17 Apr Toeplitz, Kronecker; Kolmogorov's Thm, SLLN --
Thu 19 Apr SLLN; characteristic Functions --
Fri 20 Apr Characteristic Functions HW5 (sol. on Bb.)
Tue 24 Apr Inversion formula --
Thu 26 Apr Inversion formula, weak convergence (Prokhorov's Thm) --
Fri 27 Apr Ex. class (Characteristic functions) --
Tue 1 May Weak Convergence (Prokhorov's Thm) --
Thu 3 May Weak Convergence (Continuity Lemma) --
Fri 4 May WLLN, CLT, comments HW6 (sol. on Bb.)