Nuprl Lemma : strong-law-of-large-numbers
∀p:FinProbSpace. ∀f:ℕ ⟶ ℕ. ∀X:n:ℕ ⟶ RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n])
⇒ (∀mean:ℚ
nullset(p;λs.∃q:ℚ. (0 < q ∧ (∀n:ℕ. ∃m:ℕ. (n < m ∧ (q ≤ |Σ0 ≤ i < m. (1/m) * (X[i] s) - mean|)))))
supposing E(f[0];X[0]) = mean ∈ ℚ))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
rv-iid: rv-iid(p;n.f[n];i.X[i]),
nullset: nullset(p;S),
expectation: E(n;F),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qsum: Σa ≤ j < b. E[j],
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
p-outcome: Outcome,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
rv-const: a,
rv-add: X + Y,
qsub: r - s,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
slln-lemma4,
rv-add_wf,
nat_wf,
rv-const_wf,
qmul_wf,
int-subtype-rationals,
rv-iid-add-const,
equal_wf,
squash_wf,
true_wf,
expectation-rv-add,
iff_weakening_equal,
rationals_wf,
expectation_wf,
false_wf,
le_wf,
rv-iid_wf,
random-variable_wf,
finite-prob-space_wf,
equal-wf-T-base,
qadd_wf,
expectation-rv-const,
qinverse_q,
nullset-monotone,
exists_wf,
qless_wf,
all_wf,
less_than_wf,
qle_wf,
qabs_wf,
qsum_wf,
qdiv_wf,
subtype_rel_set,
int_nzero-rational,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
p-outcome_wf,
int_seg_wf,
length_wf,
qsub_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
prod_sum_l_q,
sum_plus_q,
qsum-const,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel4,
qmul_assoc
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
axiomEquality,
rename,
sqequalRule,
lambdaEquality,
isectElimination,
applyEquality,
functionExtensionality,
minusEquality,
natural_numberEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
functionEquality,
hyp_replacement,
applyLambdaEquality,
productEquality,
setElimination,
intEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
unionElimination
Latex:
\mforall{}p:FinProbSpace. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n])
{}\mRightarrow{} (\mforall{}mean:\mBbbQ{}
nullset(p;\mlambda{}s.\mexists{}q:\mBbbQ{}
(0 < q
\mwedge{} (\mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}. (n < m \mwedge{} (q \mleq{} |\mSigma{}0 \mleq{} i < m. (1/m) * (X[i] s) - mean|)))))
supposing E(f[0];X[0]) = mean))
Date html generated:
2018_05_22-AM-00_43_15
Last ObjectModification:
2017_07_26-PM-07_01_00
Theory : randomness
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