Large Deviations techniques for performance analysis

Tentative plan

1. Probabilities under the normal curve.
2. Binomial distribution A coin tossing experiment (APG Dublin)
3. Approximating functions: Limits, asymptotic expansions, rought asymptotics
4. Computing large factorials (Stirling's Formula)
5. Rate functions: Normal, Binomial (Entropy), Cramer's theorem
6. Moment generating functions. Averages of independent observations:
7. Examples for Cramer's variational formula.
8. Application: Testing the hypothesis about the proportion H₀: p=p₀ versus H_a: p=p₁
9. Application: simple investment model
10. Application: risk management
11. Info: Exact computations, numerical methods, and Simulations (Monte Carlo)
12. Application: Simulating unlikely events
13. Gartner's Theorem for sample averages. Examples: LD for Poisson, normal, Gamma distributions.
14. Application: long queues
15. Info: Pentium processor, and long division
16. Application: Testing simple hypothesis H₀: f=f₀ versus H_a: f=f₁ (Stein's Lemma)
17. Application: multiplexing
18. Finite alphabets (empirical measures, entropy, examples)
19. Finite alphabets (Sanov's Theorem)
20. Application: Conditional limit theorems, Gibbs distributions
21. Application: Shanon's Theorems: equipartition, rate distortion
22. Random curves. Probabilities of paths
23. Application: ruin probabilities, insurance premiums
24. Application: the changepoint problem (Bucklew Ch IV E.)
25. Overview: differential equations with small noise
26. Overview: exit problems and their uses
27. Overview: long segments in a string
28. Overview: Rollback in parallel algorithms
29. Info: Refined approximation - Bahadur-Rao Theorem
30. Info: Refined approximation - Berry-Essen expansion

Theory

Stirling's Formula, binomial probabilities
Empirical measures, entropy
Combinatorial techniques
Large deviations for averages, moment generating functions
Varadhan's integrals
Large deviations for trajectories (?)
Finite State Markov chains (?)
Exact asymptotics [Bahadur-Rao theorem] (?)
Saddle point approximations (?)

Applications
		An Application to Risk-Theory
		Gibbs distributions
		Financial models: value of a risky investment
		Comparative analysis of DNA sequences 
		Analysis of computer search algorithms
		Lenghts of long queues 
		Parallell buffering
		Multiplexing
		Conditional limits and simulations
		Hypothesis testing: Chernoff bounds
		Coding: Rate distortion theory, equipartition