Martingales

Admistrivia

• Questions from the homework?
• read pp 60-61 and 87 - 93.

Martingales: background

• We can define an interesting class of processes simply by using conditional expectation:
  • E(X₂|X₁) = X₁)
  • E(X₃|X₁,X₂) = X₂)
  • E(X₄|X₁,X₂,X₃) = X₃)
  • ...
• Called a martingale
• Amazingly useful. It is a good model of:
  • betting
  • stock market
  • Random walk
  • accumulated noise in many systems
• Amazingly mathematical. You can prove truely powerful theorems:
  • Very controlled: names comes from the French for "horse bridle"
  • Eg: Suppose Var(X_t) is bounded, then X_infinity exists
  • Accurate approximations: i.e. central limit theorems
  • Lots of nice representations
  • General setting for proving "impossiblity of betting systems"
  • All very French! (They do love their theory)

Conditional expectation one more time

• Show E(X_t|X_i > lambda) > lambda a.s.
• Say what the a.s. means
• Show E(X_t|X₁ < lambda,...,X_i > lambda) > lambda a.s.

Maximal inequalities for regular random variables

• state theorem for martingales
• "prove" via though experiment (no free lunch)
• State result for general collection of n random variables
• "prove" also via no free lunch
• Now prove second claim by first proving Markov and Bonferonni inequality
• Background theorem: Markov inequality.
  • X >= 0. (You should konw what this means!)
  • P(X > k) < E(X)/k.
  • Proof: obvious!
  • X > k I_{X > k}. So, E(X) > k P(X > k). q.e.d.
• Obvious extension. Suppose E(X₀) = 1 --> P(X_n > k) < 1/k.