Advanced Stochastics

Path: math-researcher-teacher/domains/stochastics/advanced-stochastics/

🔬 Core Topics

1. Random Matrix Theory (RMT)

• •Concepts: Wigner Matrices, Semicircle Law, Marchenko-Pastur Law, Universality.
• •Script: scripts/rmt_sim.py
  • •Usage: Simulates eigenvalues of large symmetric random matrices.
  • •Output: Comparison plot of empirical eigenvalues vs Wigner's Semicircle.

2. Large Deviations Principle (LDP)

• •Concepts: Rate functions, Cramér's Theorem, Sanov's Theorem, Varadhan's Lemma.
• •Script: scripts/ldp_sim.py
  • •Usage: Simulates sample means of i.i.d. variables.
  • •Output: Visualization of probability concentration and empirical rate function approximation.

💻 How to use Scripts

Simulate Random Matrix Eigenvalues:

python scripts/rmt_sim.py

Check wigner_semicircle.png for the result.

Simulate Large Deviations:

python scripts/ldp_sim.py

Check ldp_simulation.png for the result.