I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 A. Zero-or-one laws.
 B. Optional random variable. Stopping time.
 C. Recurrence of random walk.
 D. Fine structure of stopping time.
 E. Maximal value of random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Zero-or-one laws.

roposition

(Kolmogorov zero-or-one law). For an independent process, each remote event has probability zero or one.

Proof

Let be a remote event and let . We introduce the conditional probability Let . Then by remoteness of and independence of we have Hence, Thus coincides with on . Consequently, see the proposition ( Random walk space approximation ), coincides with on . By construction of , We return to the formula with : We set and conclude

Proposition

(Preservation of stationary measure) For a stationary independent process and any we have

Proof

The is the set moved by one position to the right in the coordinate representation. But by stationarity and independence the same measure is assigned to all positions.

Proposition

(Hewitt and Savage zero-or-one law) For a stationary independent process, each permutable set has probability zero or one.

Proof

Let be a permutable set. By the proposition ( Random walk space approximation ) there is a sequence such that . For every there is a permutation such that . By independence, and by stationarity, By -additivity of , (see the proposition ( Continuity lemma )) Hence, we pass the formula to the limit and obtain

Proposition

("Infinitely often" zero-or-one law) Let be a sequence of subsets of and be a random walk. Then the set is permutable and is equal to zero or one.

Proof

For any the r.v. is invariant with respect to any permutation . Hence, the set is invariant to such permutation . Since the is decreasing as then set is invariant to any permutation for any . The statement then follows from the proposition ( Hewitt and Savage zero-or-one law ).

 Notation. Index. Contents.