S01 · implies
Before
P(a)=1/3,\ P(b)=2/3
After
P(\Omega)=1,\quad X_n,X:\Omega\to\mathbb R
Conditions: none
Rule: work · Uses: G1, work
Leaf basis: work
All variables have the same two-atom carrier and law.
A universal tail certificate from eventual equality
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derivation-mko-convergence-in-probability-01 · ocme-new142-reader-adoption-20261021
Goal and scope
Prove convergence in probability on a common finite space.
The prefix may be nonmonotone; proof comes from a universal tail formula.
Givens and symbols
work
G1— Omega={a,b}, with masses 1/3,2/3 and target X=0. X1=(2,-1), X2=(1,4), and for every n≥3 set X_n=(0,0).
ε,δ— arbitrary positive tolerancesN— tail index
Rules and sources
mko_definition· cited — All X_n and X must belong to the same probability space. Retain the tail obligations: for every epsilon>0 and every delta>0 there exists N such that every n≥N has deviation probability below delta. This slice uses an explicitly declared c/(n+s) deviation-probability profile on a valid probability domain, or an eventual-equality certificate. A finite prefix does not establish convergence. · mko-convergence-in-probabilitywork· prerequisite — Convergence in probability means controlling every n≥N for arbitrary epsilon>0 and delta>0; the empty event has probability zero.
Main derivation
S02 · implies
Before
|X_1-X|=(2,1),\quad\varepsilon=1
After
P(|X_1-X|>1)=1/3
Conditions: none
Rule: work · Uses: G1, S01, work
Leaf basis: work
The strict deviation event contains only a.
S03 · implies
Before
|X_2-X|=(1,4)
After
P(|X_2-X|>1)=2/3
Conditions: none
Rule: work · Uses: G1, S02, work
Leaf basis: work
The second deviation probability increases, without refuting a tail limit.
S04 · implies
Before
n\ge3
After
X_n(\omega)-X(\omega)=0\quad(\omega\in\Omega)
Conditions: none
Rule: work · Uses: G1, S03, work
Leaf basis: work
The declared universal formula, not two observations, gives eventual equality.
S05 · implies
Before
\varepsilon>0,\quad n\ge3
After
\{|X_n-X|>\varepsilon\}=\varnothingConditions: none
Rule: work · Uses: G1, S04, work
Leaf basis: work
Zero error exceeds no positive epsilon.
S06 · implies
Before
n\ge3,\quad\varepsilon>0
After
P(|X_n-X|>\varepsilon)=0
Conditions: none
Rule: work · Uses: G1, S05, work
Leaf basis: work
The empty event has probability zero for every tail index.
S07 · implies
Before
\varepsilon>0,\ \delta>0
After
N=3\implies\forall n\ge N:\ P(|X_n-X|>\varepsilon)=0<\delta
Conditions: none
Rule: work · Uses: G1, S06, work
Leaf basis: work
The same N=3 meets the universal tail obligation for arbitrary epsilon and delta.
S08 · implies
Before
\forall\varepsilon>0\ \forall\delta>0\ \exists N\ \forall n\ge N
After
X_n\xrightarrow{P}X=0Conditions: none
Rule: work · Uses: G1, S07, work
Leaf basis: work
The exact quantifier order proves convergence in probability.
Substeps and lemmas
No substeps in this guide.
Reverse audits
S08· premise_audit · G1, S07, work — This certificate is stronger, but the stated conclusion uses no undeclared mode-implication theorem.S07· premise_audit · G1, S06, work — Choose arbitrary epsilon and delta before supplying N.S06· premise_audit · G1, S05, work — This is not a lookup for one n.S05· premise_audit · G1, S04, work — Epsilon is arbitrary positive, not merely the tested value 1.S04· premise_audit · G1, S03, work — The quantifier n≥3 covers the infinite tail.S03· premise_audit · G1, S02, work — Prefix monotonicity is not part of the definition.S02· premise_audit · G1, S01, work — The deviation at b equals 1 and is not greater than 1.S01· premise_audit · G1, work — The space is not changed with n.
Error controls and negative controls
Strict event
Keep>epsilon and distinguish equality at 1.
Tail source
The universal n≥3 formula is evidence; the prefix is only an audit.
- The increased second probability disproves convergence. →
reject— A finite prefix does not determine the tail. - Checking epsilon=1 suffices. →
reject— The definition requires every positive epsilon. - One large n suffices. →
reject— Every n≥N must be controlled.
Result and limitations
Despite an increasing prefix deviation probability, X_n converges in probability to 0.
Verification and source binding
2 declared checks for this guide; only the listed expressions are covered.
- source commit:
b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256:
ad6c0a4edec6e457a60c1142e494599b7c1c21a43a35d0576b5fb99866768bb6 - original source SHA-256:
b0e2eaaf1242436e89c41152d920ec68ff2a8ba30a6bfe51f2b2f7c9c8ee9196 - Lean: NOT_RUN · independent AI:
SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW
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