martingale_check

std.stats.martingale_check · Level L2

The martingale conditions for a process X_0 … X_T (one row per time) on a finite probability space with a filtration given by partitions L (one row per time): whether L is a filtration; whether X is adapted (each X_t constant on the blocks of F_t); and the gaps E[X_{t+1} | F_t] − X_t for t < T, which are all 0 for a martingale. Integrability is automatic on a finite space. A martingale need not have independent increments, be Markov, or converge.

E[X_{t+1} | F_t] = X_t

Signature

martingale_check(L: f64[T, n], X: f64[T, n], p: f64[n]) → f64[], f64[], f64[T−1, n]

Requires: T ≥ 2

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size with T ≥ 2.
Equal to the reference block by block at every time, in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 575 float64 results inside the running error bound; the closest uses 44% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 41% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 87%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 29

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

is_filtration martingale_step

Called by

—

sha256:934c545adb4d2b1f34b3e03adfaac3a4e875c6e495c95597814f887abacd44c6

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.stats.martingale_check · Ω:martingale_check
Pins: sha256:30b8f5c3f106f2fb7b3c2946136344ffaf8fbc3dc8755f395c88f934514a842ethis graph and the 3 it reaches through calls
Evidence: sha256:552ca3eac8c8a0e9212e744277eb300f845a06ad5bec59e3ebc5022d69f1da3athe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.