Lecture 3 : Concentration and Correlation

Transcription

1 Lectue 3 : Cocetatio ad Coelatio 1. Talagad s iequality 2. Covegece i distibutio 3. Coelatio iequalities

2 1. Talagad s iequality Cetifiable fuctios Let g : R N be a fuctio. The a fuctio f : 1 2 Ω Ω L Ω R is said to be g-cetifiable if fo evey eal umbe s, the iequality f x s ca be checked by evealig a set of gs co-odiates of x.

3 Examples Let f : Ω1 Ω2 L ΩN R be the umbe of edge-disjoit tiagles i a adom gaph G,. Note that p N = Ω i = { e i is peset, e i is ot peset}. 2 The f has the followig two popeties: f is 1-Lipschitz ude edge exposue f is g s = 3 s cetifiable

4 Examples Let f : 1 2 Ω Ω L Ω R be the legth of a logest mootoe subsequece of a sequece of adom eal umbes betwee 0 ad 1. Note that Ω = [0,1 ]. The f has the followig two popeties: f is 1-Lipschitz ude appopiate exposue f is g s = s cetifiable i

5 Talagad s Isopeimetic Iequality Let f : 1 2 Ω Ω L Ω R be a fuctio o a poduct pobability space with mea μ. Suppose that f is g-cetifiable ad k-lipschitz. The fo some positive costat c, P f μ > c t kg μ < 2e 1 4 t 2

6 Example logest mootoe subsequece Let f : 1 2 Ω Ω L Ω R be the legth of a logest mootoe subsequece of a sequece of adom eal umbes betwee 0 ad 1. We saw f is 1-Lipschitz ude appopiate exposue f is g s = s cetifiable Also obseve Edos-Szekees that μ.

7 Example logest mootoe subsequece By Talagad s iequality, thee exists a costat C such that fo ay ε > 0 C ε P f μ > εμ < e. 2

8 Factoial Momets Covegece i distibutio Aothe way to show cocetatio is to compute the factoial momets E X = E X X 1 X 2 L X + 1 Combiatoially, factoial momets ae easie to detemie tha the usual momets E XX. They ae ameable to computatios with the pobability geeatig fuctio G X X t = E t whee t > 0

9 Example isolated vetices i a adom gaph Suppose X is the umbe of isolated vetices i G, p. The X is the umbe of choices of odeed isolated vetices i G,. This is easily detemied p E X = 1 p + 2

10 Example isolated vetices i a adom gaph E X = 1 p + 2

11 Factoial Momets It ca be show that if fo some λ R ad evey Z + E X λ as d the X X ~ POIλ. Recall that if X ~ POIλ the P X = x = e λ λ k! k

12 Example isolated vetices i adom gaphs Let X be the umbe of isolated vetices i a adom gaph p G, whee c p = log. The c c e X E + = 1 1 log Futhemoe 1 c c e X E + + = log

13 Example isolated vetices i adom gaphs Theefoe P X = x e e c e cx e x! I paticula P X = 0 e e c c

14 Factoial Momets Lévy s cotiuity it theoem implies that t if X, N ad X ae adom vaiables ad lim G t = G t X X the X coveges i distibutio to X. We ca use this to pove covegece to the Poisso distibutio whe E X λ.

15 Cetal Limit Theoem Let X N be a sequece of idepedet adom vaiables with mea μ ad stadad deviatio σ. The S σ μ d Z whee P Z z = 1 t 2 z μ 2 e 2 σ 2πσ dt

16 3. Coelatio Let Ω, F, P be a pobability bili space. Evets A1, A2, K, A F ae positively coelated if I A i P P A i S i A S i fo all S { 1, 2,..., } ad egatively coelated if fo all S { 1, 2,..., }. I A i S i i P A S i P

17 Hais-Kleitma A dowset i a patially odeed d set is a set A such hthat tx A ad y < x implies y A. A upset i a patially odeed set is a set A such that x A ad y > x implies y A. Coside the atual poduct pobability space o the Boolea lattice Q N defied by P x = p i 1 p Fo x QN let A x deote the evet x is chose the pobability of this evet is give above. i x i i x i

18 Hais-Kleitma Fo example, G is pecisely the pobability bilit space Q N whee G, p N = 2 ad the evets A x coespod to cetai subgaphs appeaig fo example a tiagle, o a Hamiltoia cycle, o a spaig biay tee. The evets A x ae upsets.

19 Hais-Kleitma THEOREM If A, B ae of the same type the they ae positively coelated, othewise they ae egatively coelated.

20 Jaso s Iequality Let Q N deote the poduct pobability space o the boolea cube ad let X QN. Let Ax be the evet that x is chose. THEOREM Let A x : x X be the evets give above ad defie μ to be the umbe of these evets which occu ad The 4 = x, y P A x A y 0 P A P x X x I x X A x e μ + Δ 2

21 Appeaace of subgaphs i adom gaphs If P is a popety of gaphs, the a theshold fo P is a fuctio τ such that P G, p P 0 if p / τ 0 P G, P 1 if p / τ 1 p Fo example we used factoial momets to show that a theshold fo havig a isolated vetex is τ = log

22 Appeaace of subgaphs i adom gaphs Fo a gaph H, lt let d H e F = max. F F H THEOREM A theshold fo the appeaace of H i G, is p 1 d H.

23 Poblem C a Compute the expected umbe μ of cycles of legth fou i the adom gaph G,. c b Compute the value of fo pat a, whee is defied i Jaso s Iequality. c Detemie Iequality. lim P G, c has o 4 - cycle usig Jaso s