Nuprl Lemma : slln-lemma4
∀p:FinProbSpace. ∀f:ℕ ⟶ ℕ. ∀X:n:ℕ ⟶ RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n])
⇒ nullset(p;λs.∃q:ℚ. (0 < q ∧ (∀n:ℕ. ∃m:ℕ. (n < m ∧ (q ≤ |Σ0 ≤ i < m. (1/m) * (X[i] s)|)))))
supposing E(f[0];X[0]) = 0 ∈ ℚ)
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,
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,
rv-iid: rv-iid(p;n.f[n];i.X[i]),
and: P ∧ Q,
rv-identically-distributed: rv-identically-distributed(p;n.f[n];i.X[i]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
cand: A c∧ B,
guard: {T}
Lemmas referenced :
slln-lemma3,
expectation_wf,
nat_wf,
false_wf,
le_wf,
rv-compose_wf,
qmul_wf,
rationals_wf,
int_seg_wf,
equal-wf-T-base,
rv-iid_wf,
random-variable_wf,
finite-prob-space_wf,
and_wf,
equal_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
axiomEquality,
rename,
productElimination,
isectElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
lambdaEquality,
independent_isectElimination,
setElimination,
independent_functionElimination,
baseClosed,
functionEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
setEquality
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{} 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)|)))))
supposing E(f[0];X[0]) = 0)
Date html generated:
2016_10_26-AM-06_54_56
Last ObjectModification:
2016_07_12-AM-08_06_33
Theory : randomness
Home
Index