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Catalog of articles in probability theory

  • 15 min read
  • 4 chapters

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Through Probability distribution, Random variable and Sample space

Core probability: selected topics

Probability theory

Basic notions (bsc)

Instructive examples (paradoxes) (iex)

  • Berkson's paradox / (2:B)
  • Bertrand's box paradox / (F:B)
  • Borel–Kolmogorov paradox / cnd (2:CM)
  • Boy or Girl paradox / (2:B)
  • Exchange paradox / (2:D)
  • Intransitive dice
  • Monty Hall problem / (F:B)
  • Necktie paradox
  • Simpson's paradox
  • Sleeping Beauty problem
  • St. Petersburg paradox / mnt (1:D)
  • Three Prisoners problem
  • Two envelopes problem

Moments (mnt)

Inequalities (inq)

  • Chebyshev's inequality / (1:R)
  • An inequality on location and scale parameters / (1:R)
  • Azuma's inequality / (F:BR)
  • Bennett's inequality / (F:R)
  • Bernstein inequalities / (F:R)
  • Bhatia–Davis inequality
  • Chernoff bound / (F:B)
  • Doob's martingale inequality / (FU:R)
  • Dudley's theorem / Gau
  • Entropy power inequality
  • Etemadi's inequality / (F:R)
  • Gauss's inequality
  • Hoeffding's inequality / (F:R)
  • Khintchine inequality / (F:B)
  • Kolmogorov's inequality / (F:R)
  • Marcinkiewicz–Zygmund inequality / mnt
  • Markov's inequality / (1:R)
  • McDiarmid's inequality
  • Multidimensional Chebyshev's inequality
  • Paley–Zygmund inequality / (1:R)
  • Pinsker's inequality / (2:R)
  • Vysochanskiï–Petunin inequality / (1:C)

Markov chains, processes, fields, networks (Mar)

  • Markov chain / (FLSU:D)
  • Additive Markov chain
  • Bayesian network / Bay
  • Birth–death process / (U:D)
  • CIR process / scl
  • Chapman–Kolmogorov equation / (F:DC)
  • Cheeger bound / (L:D)
  • Conductance
  • Contact process
  • Continuous-time Markov process / (U:D)
  • Detailed balance / (F:D)
  • Examples of Markov chains / (FL:D)
  • Feller process / (U:G)
  • Fokker–Planck equation / scl anl
  • Foster's theorem / (L:D)
  • Gauss–Markov process / Gau
  • Geometric Brownian motion / scl
  • Hammersley–Clifford theorem / (F:C)
  • Harris chain / (L:DC)
  • Hidden Markov model / (F:D)
  • Hidden Markov random field
  • Hunt process / (U:R)
  • Kalman filter / (F:C)
  • Kolmogorov backward equation / scl
  • Kolmogorov's criterion / (F:D)
  • Kolmogorov's generalized criterion / (U:D)
  • Krylov–Bogolyubov theorem / anl
  • Lumpability
  • Markov additive process
  • Markov blanket / Bay
  • Markov chain mixing time / (L:D)
  • Markov decision process
  • Markov information source
  • Markov kernel
  • Markov logic network
  • Markov network
  • Markov process / (U:D)
  • Markov property / (F:D)
  • Markov random field
  • Master equation / phs (U:D)
  • Milstein method / scl
  • Moran process
  • Ornstein–Uhlenbeck process / Gau scl
  • Partially observable Markov decision process
  • Product-form solution / spr
  • Quantum Markov chain / phs
  • Semi-Markov process
  • Stochastic matrix / anl
  • Telegraph process / (U:B)
  • Variable-order Markov model
  • Wiener process / Gau scl

Gaussian random variables, vectors, functions (Gau)

Conditioning (cnd)

Specific distributions (spd)

Empirical measure (emm)

Limit theorems (lmt)

Large deviations (lrd)

  • Large deviations theory
  • Contraction principle
  • Cramér's theorem
  • Exponentially equivalent measures
  • Freidlin–Wentzell theorem
  • Laplace principle
  • Large deviations of Gaussian random functions / Gau
  • Rate function
  • Schilder's theorem / Gau
  • Tilted large deviation principle
  • Varadhan's lemma

Random graphs (rgr)

Random matrices (rmt)

Stochastic calculus (scl)

  • Itô calculus
  • Bessel process
  • CIR process / Mar
  • Doléans-Dade exponential
  • Dynkin's formula
  • Euler–Maruyama method
  • Feynman–Kac formula
  • Filtering problem
  • Fokker–Planck equation / Mar anl
  • Geometric Brownian motion / Mar
  • Girsanov theorem
  • Green measure
  • Heston model / fnc
  • Hörmander's condition / anl
  • Infinitesimal generator
  • Itô's lemma
  • Itô calculus
  • Itô diffusion
  • Itô isometry
  • Itô's lemma
  • Kolmogorov backward equation / Mar
  • Local time
  • Milstein method / Mar
  • Novikov's condition
  • Ornstein–Uhlenbeck process / Gau Mar
  • Quadratic variation
  • Random dynamical system / rds
  • Reversible diffusion
  • Runge–Kutta method
  • Russo–Vallois integral
  • Schramm–Loewner evolution
  • Semimartingale
  • Stochastic calculus
  • Stochastic differential equation
  • Stochastic processes and boundary value problems / anl
  • Stratonovich integral
  • Tanaka equation
  • Tanaka's formula
  • Wiener process / Gau Mar
  • Wiener sausage

Malliavin calculus (Mal)

  • Malliavin calculus
  • Clark–Ocone theorem
  • H-derivative
  • Integral representation theorem for classical Wiener space
  • Integration by parts operator
  • Malliavin derivative
  • Malliavin's absolute continuity lemma
  • Ornstein–Uhlenbeck operator
  • Skorokhod integral

Random dynamical systems (rds)

Random dynamical system / scl

  • Absorbing set
  • Base flow
  • Pullback attractor

Analytic aspects (including measure theoretic) (anl)

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