Mathematics / advanced
Concentration and nonasymptotic probability
Control finite-sample fluctuations with exponential moments, martingale bounds, isoperimetry, and random-matrix estimates.
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Inside this module
Module 01 advanced
Concentration and nonasymptotic probability
Control finite-sample fluctuations with exponential moments, martingale bounds, isoperimetry, and random-matrix estimates.
- 01
Exponential moments and the Chernoff method
Convert exponential moments into directed tail bounds and optimize only over the moment-generating function's valid domain.
29 min - 02
Hoeffding's lemma and bounded sums
Use deterministic interval widths to obtain finite-sample Gaussian-shaped tails for independent bounded variables.
29 min - 03
Bernstein, Bennett, and subexponential tails
Use variance and one-sided increment bounds, distinguish exact constants, and specify the local-mgf convention for subexponential variables.
33 min - 04
Subgaussian variables and Orlicz norms
Separate a centered mgf variance proxy from an uncentered Orlicz norm and compute the constants in elementary examples.
31 min - 05
Bounded differences and McDiarmid's inequality
Control a nonlinear function of independent inputs through deterministic coordinate sensitivities and recognize when dependence defeats the argument.
30 min - 06
Martingale Bernstein and Freedman's inequality
Replace independent-sum variance by predictable conditional variance and control crossings with a deterministic variance budget.
33 min - 07
Gaussian concentration and Lipschitz functions
Control nonlinear observables of a Gaussian vector with the correct Euclidean Lipschitz scale and covariance normalization.
30 min - 08
Log-Sobolev inequalities and Herbst's argument
Derive a centered exponential-moment bound from a precisely normalized entropy–energy inequality and justify the regularity steps.
34 min - 09
Symmetrization and Rademacher processes
Bound expected empirical-process suprema using independent signs while retaining measurability, envelope, absolute-value, and normalization conventions.
33 min - 10
Matrix concentration and operator-norm bounds
Apply Hermitian matrix Bernstein with the correct dimension and variance parameters, and distinguish eigenvalue tails from singular-value norms.
34 min