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Brun’s Sieve

Brun’s Sieve Let B 1 , …, B m be events, X i the indicator random variable for Bi and X = X 1 + … + X m the number of Bi that hold. Let there be a hidden parameter n (so that actually m = m ( n ), B i = B i ( n ), X = X( n )) which will define the following o , O notation.

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Brun’s Sieve

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  1. Brun’s Sieve • Let B1, …, Bm be events, Xi the indicator random variable for Bi and X = X1 + … + Xm the number of Bi that hold. • Let there be a hidden parameter n (so that actually m = m(n), Bi = Bi(n), X = X(n)) which will define the following o, O notation. • Define S(r) = ∑ Pr[Bi1Λ…ΛBir ], the sum over all sets {i1,…,ir}  {1,…,m}.

  2. Theorem 8.3.1 Suppose there is a constant μso that E[X] = S(1) → μ and such that for every fixed r, E[X(r) / r!] = S(r) →μr/ r!. Then Pr[X = 0] → and indeed for every t Pr[X = t] →

  3. Pr[X = r] ≤ S(r) = ∑ Pr[ ], where {i1,…,ir}  {1,…,m}. • The Inclusion-Exclusion Principle gives that Pr[X = 0] = Pr[ ] = 1 – S(1) + S(2) -…+(-1)rS(r)… • Bonferroni’s inequality: Let P(Ei) be the probability that Ei is true, and be the probability that at least one of Ei,…, En is true. Then

  4. Proof. We do only the case t = 0. • Fix є> 0. Choose s so that • The Bonferroni Inequalities states that, in general, the inclusion-exclusion formula alternatively over and underestimates Pr[X = 0]. In particular, • Select no(the hidden variable) so that for n  no, for 0 ≤ r ≤ 2s.

  5. Proof(cont.) • For such n Pr[X = 0] ≤ + є • Similarly, taking the sum to 2s+1 we find no so that for n  no Pr[X = 0] ≤ - є • As є wasarbitrary Pr[X = 0] →

  6. Let G~ G(n,p), the random graph and let EPIT represent the statement that every vertex lies in a triangle. • Theorem 8.3.2 Let c > 0 be fixed and let p = p(n),μ=μ(n) satisfy p3 = μ, = Then Pr[G(n,p) |= EPIT] =

  7. Proof. • First fix xV(G). • For each unordered y, z V(G) – {x} let Bxyz be the event that {x,y,z} is a triangle of G. • Let Cx be the event and Xx be the corresponding indicator random variable. • We use Janson’s Inequality to bound E[Xx] = Pr[Cx]. Here p = o(1) so є = o(1). as defined above.

  8. Proof(cont.) • Dependency xyz ~ xuv occurs if and only if the sets overlap (other than x). Hence • Since . Thus • Now define the number of vertices x not lying in a triangle. Then from Linearity of Expectation,

  9. Proof(cont.) • We need to show that the Poisson Paradigm applies to X. Fix r. Then the sum over all sets of vertices {x1,…,xr}. All r-sets look alike so where x1,…,xr are some particular vertices. But the conjunction over 1 ≤ i ≤ r and all y,z.

  10. Proof(cont.) • We apply Janson’s Inequality to this conjunction. Again є = p3 = o(1). • The number of {xi,y,z} is , the overcount coming from those triangles containing two(or three of the xi). (Here it is crucial that r is fixed.) Thus As before Δ is p5 times the number of pairs xiyz~ xjy’z’. There are O(rn3) = O(n3) terms with i=j and O(r2n2) = O(n2) terms with ij so again Δ = o(1). Therefore and

  11. Large Deviations • Given a point in the probability space(i.e., a selection of R) we call an index set J I a disjoint family (abbreviated disfam) if • Bjfor every jJ. • For no j, j’ J is j ~ j’. If, in addition, • If j’J and Bj’ thenj ~ j’ for some j J. Then we call J a maximal disjoint family (abbreviated maxdisfam).

  12. Lemma 8.4.1 With the above notation and for any integer s, Pr[there exists a disfam J, |J| = s] ≤ • Proof. • Let denote the sum over all s-sets JI with no j ~ j’. • Let denote the sum over ordered s-tuples (j1 ,…, js) with {j1 ,…, js} forming such a J. • Let denote the sum over all ordered s-tuples (j1,…, js).

  13. Proof(cont.) • Pr[there exists a disfam J, |J| = s]

  14. For smaller s we look at the further condition of J being a maxidisfam. To that end we let μs denote the minimum, over all j1, … , js of ,the sum taken over all iI except those i with i ~ jl for some 1≤l≤s. • In application s will be small (otherwise we use Lemma 8.4.1) and μs will be close to μ. For some applications it is convenient to set and note that μs >= μ – sv.

  15. Lemma 8.4.2 With the above notation and for any integer s, Pr[there exists a maxdisfam J, |J| = s] ≤ • Proof. • As in Lemma 8.4.1 we bound this probability by of J = {j1, … , js} being a maxdisfam. For this to occur J must first be a disfam and then , where is the conjunction over all i I except those with i ~ jl for some 1 ≤l ≤ s.

  16. Proof(cont.) • We apply Janson’s Inequality to give an upper bound to .The associated values satisfy the latter since has simply fewer addends. Thus and

  17. When Δ = o(1) and vμ = o(1) or, more generally, μ3μ = μ + o(1), then Lemma 8.4.2 gives a close approximation to the Poisson Distribution since Pr[there exists a maxdisfam J, |J| = s] For s≤ 3μ and the probability is quite small for larger s by Lemma 8.4.1

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