Alphabeta MathProbability
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
1 pageThis page moves from measure spaces of total mass 1 to the standard probability toolkit built on them.
6 definitions, 6 lemmas, 7 theorems, 11 corollaries, 2 remarksExamples & counterexamples →
Level 1
2 pages- Conditional Expectation22 results
Conditional expectation is first constructed as a measurable integrable version by Radon–Nikodym and then made into an almost-sure class by uniqueness.
4 definitions, 5 lemmas, 9 theorems, 3 corollaries, 1 remarkExamples & counterexamples → This page develops independence beyond the earlier finite models.
6 definitions, 6 lemmas, 8 theorems, 6 corollariesExamples & counterexamples →
Level 2
3 pages- Discrete Time Martingales21 results
A filtration records the information available at each nonnegative integer time.
9 definitions, 3 lemmas, 5 theorems, 4 corollariesExamples & counterexamples → This page separates countable products of arbitrary probability spaces from arbitrary-index extension of consistent laws on standard-Borel coordinates.
5 definitions, 6 lemmas, 4 theorems, 4 corollaries, 2 remarksExamples & counterexamples →This page fixes the real-valued versions of almost-sure, probability, Lᵖ, and distributional convergence, proves their principal implications, and records precisely which converses fail.
5 definitions, 2 lemmas, 14 theorems, 1 corollary, 1 remarkExamples & counterexamples →
Level 3
2 pagesAn explicit real coding starts with canonical binary rows, interleaves their digits, and embeds the sequence in separated ternary cylinders.
2 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →Weak laws arise first from variance control and then from truncating large summands.
5 definitions, 10 lemmas, 9 theorems, 3 corollariesExamples & counterexamples →
Level 4
2 pagesA conditional probability for a single event is an almost-sure class.
6 definitions, 5 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →- Strong Laws of Large Numbers16 results
Truncation and variance summability establish the IID integrable strong law.
1 definition, 5 lemmas, 5 theorems, 4 corollaries, 1 remarkExamples & counterexamples →
Level 5
1 pageBounded continuous tests define weak convergence. Portmanteau supplies set criteria and mapping results; compact approximation and Prokhorov characterize tightness. The…
7 definitions, 8 lemmas, 8 theorems, 3 corollaries, 1 remarkExamples & counterexamples →
Level 6
1 pageCharacteristic functions encode a real probability law by the bounded tests x↦ eⁱᵗˣ.
2 definitions, 5 lemmas, 6 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 7
1 page- Central Limit Theorems16 results
Central limit theorems describe the laws of normalized sums. The scalar argument begins with an explicit normal transform and a second-order expansion requiring only a finite…
3 definitions, 6 lemmas, 4 theorems, 2 corollaries, 1 remarkExamples & counterexamples →
Level 8
2 pagesA Gaussian process is specified through all finite linear combinations, with singular covariance matrices and variance zero retained.
6 definitions, 7 lemmas, 6 theorems, 2 corollariesExamples & counterexamples →- Martingale Inequalities and Convergence18 results
Upcrossing counts are defined without selecting optimal crossing times.
3 definitions, 2 lemmas, 11 theorems, 2 corollariesExamples & counterexamples →
Level 9
1 page- Stopping Times and Optional Stopping17 results
Stopping times are defined by literal finite-horizon events, with boundedness kept distinct from almost-sure finiteness.
3 definitions, 5 lemmas, 5 theorems, 3 corollaries, 1 remarkExamples & counterexamples →
Level 10
2 pagesBrownian motion is placed in its raw natural filtration Natural and usual augmented Brownian filtrations, which is then completed by the terminal null sets and made right-continuous.
5 definitions, 4 lemmas, 6 theorems, 4 corollaries, 1 remarkExamples & counterexamples →- Markov Kernels and Markov Chains22 results
A transition kernel records one-step dynamics on an arbitrary measurable state space.
7 definitions, 4 lemmas, 8 theorems, 3 corollariesExamples & counterexamples →
Level 11
2 pages- Brownian Path Properties20 results
This page studies the sample paths of the Brownian motion of brownian-motion-construction-and-continuity and brownian-motion-markov-properties-and-hitting-times, as paths rather than as a Markov process or a martingale.
2 definitions, 4 lemmas, 8 theorems, 5 corollaries, 1 remarkExamples & counterexamples → This page develops countable-state, discrete-time recurrence and hitting-time theory from the transition-matrix and stopping-time conventions.
9 definitions, 5 lemmas, 8 theorems, 5 corollariesExamples & counterexamples →
Level 12
2 pagesThis page develops stationary laws for Markov chains and the limits that the stationary law governs.
5 definitions, 4 lemmas, 10 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page constructs the Ito integral with respect to Brownian motion along the standard route: continuous-time vocabulary Continuous-time adapted processes and martingales…
6 definitions, 4 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →
Level 13
1 page- Itos Formula and Brownian Martingales21 results
The page fixes the class of continuous Brownian Ito processes Continuous Brownian Ito processes and defines their quadratic covariation along deterministic partition sequences Quadratic covariation of Brownian Ito processes.
3 definitions, 2 lemmas, 8 theorems, 6 corollaries, 2 remarksExamples & counterexamples →
Level 14
1 page- Forward Itô SDE Well-Posedness5 results
This page fixes the finite-dimensional Itô SDE and strong-solution conventions, including the usual filtered-space hypotheses, finite-coordinate matrix integral, uniform…
1 definition, 2 lemmas, 2 theoremsExamples & counterexamples →