Finite-valued IID strong-law qualification and fixed-n boundaries

Goal and scope

Match the named SLLN hypotheses while separating prefix audits from a.s. proof.

Infinite IID is a premise and finite-valued SLLN is a named theorem.

Givens and symbols

• work

• G1 — On a supplied probability space, (X_i)_{i≥1} is an infinite IID sequence: every finite distinct-index joint law factors into Bernoulli(1/2) marginals. This is a whole-process premise, not inferred from finite data.

• bar X_n — sample mean
• μ — common marginal mean

Rules and sources

• mko_definition · cited — Assume an explicit infinite IID process and a normalized finite common marginal law with mean mu. The strong-law conclusion is P(lim_(n→infinity) n^(-1)sum_(i=1)^n X_i=mu)=1. Exact families recalculate the marginal mean, a fixed prefix sample mean and a complete fixed-n deviation-event probability. They expressly do not certify the probability-one limit. The infinite_iid certificate token is a declared process premise, not verification of infinite independence. Zero variance is allowed. Non-IID and nonfinite support are outside this finite-valued theorem scope, not counterexamples to every SLLN; a finite prefix does not establish a limit and a different target is not the common mean. · mko-strong-law-large-numbers-finite-valued-iid
• work · prerequisite — Explicit named theorem prerequisite: an infinite IID finite-valued sequence on a given probability space has sample means converging a.s. to the common mean; also assume the binomial law and continuity from above of probability.

Main derivation

\{0,1\},\quad p=(1/2,1/2)

p_0,p_1\ge0,\quad p_0+p_1=1

\mu=0(1/2)+1(1/2)

\mu=1/2,\quad |X_i|\le1\text{ a.s.}

x_1,x_2,x_3,x_4=(1,0,1,1)

\bar x_4=3/4

S_4\sim\operatorname{Binomial}(4,1/2)

P(S_4=k)=(1,4,6,4,1)_k/16\quad(k=0,1,2,3,4)

|S_4/4-1/2|\ge1/4

S_4\in\{0,1,3,4\},\quad P=10/16=5/8

\text{infinite IID common finite law}

P\!\left(\lim_{n\to\infty}\frac1n\sum_{i=1}^{n}X_i=1/2\right)=1\quad\text{(named SLLN)}

P(\lim_n\bar X_n=1/2)=1

\bar X_n\to1/2\quad\text{almost surely}

X_i\equiv c\text{ a.s.}\quad\text{(boundary)}

\bar X_n=c\text{ a.s. for every }n

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — Nonfinite support is outside this slice, not a claim that every general strong law fails.
• S07 · premise_audit · G1, S06, work — The all-ones event E is the intersection over m≥1 of measurable events{X1 through Xm all equal1}. Each finite-prefix event has probability 2^(-m); continuity from above gives P(E)=lim2^(-m)=0, without treating sample points as paths or singletons.
• S06 · premise_audit · G1, S05, work — The value 5/8 is not the limit-event probability; the limit uses the named theorem.
• S05 · premise_audit · G1, S04, work — k=2 is excluded while k=1,3 are included.
• S04 · premise_audit · G1, S03, work — The five lattice points exhaust the possible sums.
• S03 · premise_audit · G1, S02, work — A prefix value different from mean 1/2 is no contradiction.
• S02 · premise_audit · G1, S01, work — Boundedness concerns every i under the common law.
• S01 · premise_audit · G1, work — This alone does not construct an infinite joint process.

Error controls and negative controls

Process and marginal

Keep finite-law validation separate from the infinite IID premise.

Event layers

Do not interchange fixed-n deviation5/8 with probability 1 of a limit event.

• The prefix mean 3/4 refutes the strong law. → reject — A limit does not require equality at finite n.
• The strong law requires sigma²>0. → reject — Constant IID sequences satisfy this finite-valued strong law.
• Nonfinite support violates every SLLN. → reject — It is only outside this slice; other strong laws have different hypotheses.

Result and limitations

By the named SLLN the mean tends a.s. to 1/2; fixed-n=4 deviation probability 5/8 neither proves nor refutes that limit.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: bdda8ad449130122c4fb072fb2338389ae3e3f6902e49ee7a1976ed50d7d5044
• original source SHA-256: da6d2c78524cf0fe83d2656b7f968e993ea5d1eb189113dbdbc2977b902653d6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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