Further Topics in Probability, Spring 2016
Márton Balázs  
Email:  m.balazs@our_city.ac.countrycode  
Tel:  +44 (0)117 9287991  
Office:  Maths 3.7  
Drop in Sessions:  Tuesdays 10:30  11:30 on teaching weeks, Maths 3.7  
except for the 8th of March, which is cancelled.  
Q&A Session:  Friday 20^{th} May, 10:00am, SM2. 

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: squareroot of the average ≠ average of the squareroot.
Day  Topics  Homework due: 

Mon 25 Jan  Review of the unit, basic discrete distributions   
Tue 26 Jan  Convolution (discrete cases)   
Thu 28 Jan  Ex. Class (Neg.bin., Poi., Uniform convolutions)   
Mon 1 Feb  Normal distribution, Convolution (Gaussian)   
Tue 2 Feb  Convolution (Cauchy)   
Thu 4 Feb  Convolution (Cauchy, Exponential)  HW1 (sol. on Bb.) 
Monday's class rescheduled to Tuesday  
Tue 9 Feb  Poisson process, Chi square   
Tue 9 Feb  Generating functions (properties; examples; of convolutions)   
Thu 11 Feb  Ex. Class (Moran's example, neg.bin.+combi. generating fcts)   
Mon 15 Feb  Generating functions (random no. of summands)   
Tue 16 Feb  Generating functions (GaltonWatson process)   
Thu 18 Feb  Generating functions (Critical GW process; random walk: level 1 hitting time)  HW2 (sol. on Bb.) 
Mon 22 Feb  Generating functions (random walk)   
Tue 23 Feb  Generating functions (Weak convergence, Poisson approximation)   
Thu 25 Feb  Ex. Class (Generating functions: Catalan example, two sixes problem)   
Mon 29 Feb  Weak convergence examples, Weak Law of Large Numbers   
Mon 29 Feb  Weak Law of Large Numbers, Stirling's formula   
Tue 1 Mar  Stirling's formula, DeMoivre Laplace CLT   
Thu 3 Mar  Ex. Class (Generating functions: RW probabilities)  HW3 (sol. on Bb.) 
I'm away  
Mon 14 Mar  DeMoivre Laplace CLT   
Mon 14 Mar  Measure Theory (basic notions, probability)   
Tue 15 Mar  Measure Theory, Probabilistic tools   
Tue 15 Mar  Probabilistic tools   
Thu 17 Mar  Probabilistic inequalities; types of convergences  HW4 (sol. on Bb.) 
Easter vacation  
Mon 11 Apr  Convergences   
Tue 12 Apr  Strong Law of Large Numbers (Kolmogorov's ineq., KolmogorovKhinchin)   
Thu 14 Apr  Ex. Class (Gen. fcts; convergence types)   
Mon 18 Apr  Toeplitz, Kronecker; Kolmogorov's Thm, SLLN   
Tue 19 Apr  Characteristic Functions   
Thu 21 Apr  Characteristic Functions (Inversion formula)  HW5 (sol. on Bb.) 
Mon 25 Apr  Inversion formula and its consequences   
Tue 26 Apr  Weak Convergence, CLT (Prokhorov's Thm)   
Thu 28 Apr  Ex. Class (Characteristic functions)   
Mon 2 May  Bank holiday  
Tue 3 May  Weak Convergence, CLT (Continuity Lemma)   
Wed 4 May  Weak Convergence, WLLN, CLT   
Thu 5 May  Weak Convergence, CLT (Comments on the CLT)  HW6 (sol. on Bb.) 
If you have any questions, please contact me (see on top). Below is my schedule.
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