I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are:
- Borel-Cantelli Lemmas
- Modes of Convergence and individual properties
- Laws of Large Numbers
- Central Limit Theorem
- Levy Convergence Theorem and Kolmogorov's Maximal Inequality
- Expectation and Limit Theorems like MCT, DCT and Fatou's Lemma.
I know that there are several books offering enough theory but I am in no need of those. I would want a book containing stimulating problems because just by studying theory, I cannot be sure that I can apply those concepts in relevant areas. Thank you.
Here are three of my favourite books on probability:
1) Brownian Motion by Peter Morters and Yuval Peres
http://www.amazon.co.uk/Brownian-Cambridge-Statistical-Probabilistic-Mathematics/dp/0521760186/ref=sr_1_1?ie=UTF8&qid=1433016008&sr=8-1&keywords=brownian+motion+peter+morters
This reference is not as great for the convergence issues you requested, but it does have hints and solutions for a range of topics in probability from Markov processes, harmonic functions and Hausdorff dimension, to potential theory, conformal invariance and stochastic Loewner evolutions.
2) Schaum's Outline Probability and Statistics by Murray Spiegel and John Schiller
http://www.amazon.co.uk/Schaums-Outline-Probability-Statistics-4th/dp/007179557X/ref=sr_1_2?ie=UTF8&qid=1433015644&sr=8-2&keywords=schaum+outline+probability
This has plenty of solved problems in probability but not so much on convergence issues.
3) Probability Models by John Haigh
http://www.amazon.co.uk/Probability-Models-Springer-Undergraduate-Mathematics/dp/1852334312/ref=sr_1_3?ie=UTF8&qid=1433015782&sr=8-3&keywords=probability+john+haigh
This has solved problems for classic topics in probability theory but particularly on convergence and limit theorems.