Example
Xi~P({1})=P({-1})=1/2
St is the standardized partial sum of Xi
S0=0, stationary increments
Then St convergence in distribution to Brownian motion
CLT
Definition
increments
Stationary
Independence
Bt
Normal distribution (0, constant)
Property
Multivariate Gaussian distribution
expectation=0
Covariance=min(s,t)*sigma_square
Bt-Bs~B(t-s)
Sample paths
Continuous
Nowhere differentiable
classical integration methods failed
Unbounded variation on any finite interval
What does this mean?
irregularity
B is 1/2-self-similar
H-self-similar
Comparison vs. Poisson process
increments
sample path
Extension
Brownian motion with drift
Standard Brownian motion
Geometric Brownian motion
Gaussian White and Colored noise
Functional Central Limited Theorem
Simple version (iid)
Stochastic Process
Finite dimensional distribution converge
Uniform tightness
Stochastic compactness
MD array
