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Alternative form of Chernoff's bound #1140

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affeldt-aist merged 1 commit into from
Jan 8, 2024
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Alternative form of Chernoff's bound #1140

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6 changes: 4 additions & 2 deletions theories/probability.v
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Original file line number Diff line number Diff line change
Expand Up @@ -531,11 +531,13 @@ Qed.
Definition mmt_gen_fun (X : {RV P >-> R}) (t : R) := 'E_P[expR \o t \o* X].

Lemma chernoff (X : {RV P >-> R}) (r a : R) : (0 < r)%R ->
P [set x | X x >= a]%R * (expR (r * a))%:E <= mmt_gen_fun X r.
P [set x | X x >= a]%R <= mmt_gen_fun X r * (expR (- (r * a)))%:E.
Proof.
move=> t0; rewrite /mmt_gen_fun; have -> : expR \o r \o* X =
move=> t0.
rewrite /mmt_gen_fun; have -> : expR \o r \o* X =
(normr \o normr) \o [the {mfun T >-> R} of expR \o r \o* X].
by apply: funext => t /=; rewrite normr_id ger0_norm ?expR_ge0.
rewrite expRN lee_pdivl_mulr ?expR_gt0//.
rewrite (le_trans _ (markov _ (expR_gt0 (r * a)) _ _ _))//; last first.
exact: (monoW_in (@ger0_le_norm _)).
rewrite ger0_norm ?expR_ge0// muleC lee_pmul2l// ?lte_fin ?expR_gt0//.
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