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,
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,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ
Lemmas :
slln-lemma4,
rv-add_wf,
rv-const_wf,
qmul_wf,
int-subtype-rationals,
rv-iid-add-const,
expectation-rv-add,
iff_weakening_equal,
qadd_wf,
squash_wf,
true_wf,
rationals_wf,
expectation-rv-const,
equal-wf-T-base,
equal_wf,
expectation_wf,
false_wf,
le_wf,
rv-iid_wf,
nat_wf,
random-variable_wf,
finite-prob-space_wf,
Error :qinverse_q,
nullset-monotone,
exists_wf,
Error :qless_wf,
all_wf,
less_than_wf,
Error :qle_wf,
Error :qabs_wf,
Error :qsum_wf,
qdiv_wf,
subtype_rel_set,
Error :int_nzero-rational,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
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,
sq_stable__le,
Error :sum_plus_q,
Error :prod_sum_l_q,
Error :qsum-const,
Error :qmul_over_plus_qrng,
Error :qmul_over_minus_qrng,
Error :qmul_comm_qrng,
Error :qmul_ac_1_qrng,
Error :qmul-qdiv-cancel4,
qmul_assoc
\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:
2015_07_17-AM-08_04_23
Last ObjectModification:
2015_02_03-PM-09_46_18
Home
Index