An exact L-squared tail on a finite law

Goal and scope

Compute the second error moment and give a tail for any tolerance.

Fix p=2; no other mode is inferred from a finite prefix.

Givens and symbols

• work

• G1 — Omega={a,b} has masses 1/4,3/4; X(a)=1, X(b)=-1; for every n≥1, X_n(a)=1+2/(n+1), X_n(b)=-1-1/(n+1).

• p — fixed exponent2
• η — moment tolerance
• ε — norm tolerance

Rules and sources

• mko_definition · cited — This slice uses positive integer p, a complete finite atomic law and symbolic error amplitudes c_i/(n+s). Compute exact p-moments and epsilon tail bounds. A finite prefix is insufficient; unrestricted almost-sure, other-Lq or reverse probability-mode implications are not inferred without their additional theorem assumptions. · mko-lp-convergence
• work · prerequisite — Finite-law p-moments sum weighted absolute pth powers; the L2 norm is the square root of the second moment; L2 convergence means the L2 error norm tends to zero; floor(u)+1>u.

Main derivation

p=2,\quad1/4+3/4=1

X_n,X\in L^2(\Omega,P)\quad(n\ge1)

X_n-X

(2/(n+1),-1/(n+1))

|X_n-X|^2

(4/(n+1)^2,1/(n+1)^2)

E|X_n-X|^2

\frac14\frac4{(n+1)^2}+\frac34\frac1{(n+1)^2}=\frac7{4(n+1)^2}

\|X_n-X\|_2^2=7/[4(n+1)^2]

\|X_n-X\|_2=\sqrt7/[2(n+1)]

\eta>0,\quad N=\lfloor7/(4\eta)\rfloor+1

n\ge N\implies\frac7{4(n+1)^2}\le\frac7{4(n+1)}<\eta

\eta=\varepsilon^2\quad(\varepsilon>0)

n\ge N(\varepsilon^2)\implies\|X_n-X\|_2<\varepsilon

\forall\varepsilon>0\ \exists N\ \forall n\ge N:\|X_n-X\|_2<\varepsilon

X_n\to X\text{ in }L^2

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — No general cross-mode theorem is inferred from this calculation.
• S07 · premise_audit · G1, S06, work — Keep norm tolerance distinct from moment tolerance.
• S06 · premise_audit · G1, S05, work — The comparison of denominators uses n+1≥1.
• S05 · premise_audit · G1, S04, work — The denominator simplifies because n+1 is positive.
• S04 · premise_audit · G1, S03, work — The weighted constant is 7/4, not 4+1.
• S03 · premise_audit · G1, S02, work — The denominator must be squared too.
• S02 · premise_audit · G1, S01, work — Do not cancel b's negative error against a before squaring.
• S01 · premise_audit · G1, work — The positive integer p=2 is fixed across n.

Error controls and negative controls

Weights

The weighted constant is(1/4)4+(3/4)1=7/4.

Two tolerances

Set the moment tolerance to epsilon squared before concluding norm<epsilon.

• Square the average signed error for the L2 moment. → reject — Square atomwise before averaging.
• The norm is 7/[4(n+1)²]. → reject — That is the squared norm.
• Ten decreasing moments prove convergence. → reject — A certificate for every n≥N is required.

Result and limitations

E|X_n-X|²=7/[4(n+1)²] tends to 0, proving L2 convergence.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 2453a98088bbf8b7dbe596f3cfb3612829c1032a34024986b8c53e13854e7840
• original source SHA-256: 912ab75eb29e9ea8f780cff43d28934053dc4a26d8ca8f4f3dc9273b99eb0faa
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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