Tools from Stochastic Calculus 1
summary
Defines a multilinear extension on [-1,1]^n and builds a coordinate-revealing random walk X0→Xn: choose a random permutation Ki and iid ±1 epsilons, set Xi−Xi−1=εi eKi, so Xn is uniform on the hypercube. Shows Mi=f(Xi) is a martingale and uses orthogonality to write Var(f)=E sum (Mi−Mi−1)^2, then bounds each increment by the final jump to get Var(f) ≤ sum influences. Also sketches continuous martingale sampling schemes (absolute-value t path and Brownian stopping).
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chapters (12)
- 0:00Introduction to course
- 2:03Defining multilinear extensions
- 5:58Overview of Poincare inequality
- 12:21Fourier decomposition
- 18:51Stochastic process construction
- 23:42Martingale properties
- 27:06Variance and differences
- 32:34Proving the inequality
- 36:46Conclusion of proof
- 42:30Q&A on Poincare inequality
- 45:22Alternative sampling methods
- 51:53Continuous sampling overview
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