Proof of Theorem 4. Before going to the proof of Xxxxxxx 4, we will give a high-level intuition. The core idea is to consider the two terms of (10), and to make a distinction depending on how much freedom the distinguisher has in influencing the rekeying of the σ message-dependent evaluations of E. We consider two cases: Ð T Ð
Proof of Theorem 4. 2 In Section 4.2.1 we prove three lemmas for the probability that the empirical speed is above a given threshold. These lemmas will be used in Section 4.2.2 to prove Theo- rems 4.2(a)–(b). In Section 4.2.3 we prove Theorems 4.2(c).
Proof of Theorem 4. 4.1 It is trivial that equation (4.13)-(4.16) hold for t = T . Assuming equation (4.9) and equations (4.13)-(4.16) hold for t ≥ k + ∆t, we are now checking the case for t = k. Let u = (uk, u∗k+∆t, ..., u∗T ), from definition (4.7), (4.8) and the Tower Property, the utility function can be written as J(k, Xk, ψk; u) = E [Xu] — γ V ar [Xx] k,Xk,ψk T = Ek,X ,ψ hE 2 k,Xk,ψk T uk Xu∗ i k k k+∆t,Xk+∆t
Proof of Theorem 4. 4.2 We can determine expected wealth at time t + ∆t based on time those at t as Et,X ,ψ [Xu ] = Σ (1 + µt,i∆t + βi)ψt,i ut,i (4.39)
Proof of Theorem 4. We prove that ψ′(X1; X2; X3; ...; XmǁZ) satisfies the four properties way secret key rate from X1 to X2, ..., Xm in the presence of Z which in turn is greater than or equal to the stated lower bound (the details are suppressed). Proof of Theorem 2. Fix the probability distribution p(x1, x2, ..., xm, z) on (X1, X2, ..., Xm, Z) and assume that (X1, X2, ..., Xm, Z) take values in the discrete finite sets ∆i, of Theorem 2. We prove this in two stages: First we assume that C.H.{ψ} satisfies the last three properties and prove that ψ′(X1; X2; X3; ...; XmǁZ) satisfies all four properties; and then we prove that C.H.{ψ} satisfies the last three properties. Property number 1: ψ′(X1; X2; X3; ...; XmǁZ) =
Proof of Theorem 4. We prove that ψ′(X1; X2; X3; ...; Xm Z) satisfies the four properties
Proof of Theorem 4. 3.1. By Theorem 4.2.1 we have (up to subsequences) ǁµ(n) − µ(∞)ǁL2 = op(1) and ǁµ(n)ǁHs = Op(1). By Xxxxxx’x Theorem and the fact that f∞ is quadratic we have f∞(µ(n)) = f∞ (µ(∞)) + ∂f∞ .µ(∞); µ(n) − µ(∞)Σ + 1 ∂2f 2 ∞ .µ(∞); µ(n) − µ(∞)Σ . Since µ(∞) minimizes f∞ the linear term above must be zero. Hence f∞(µ(n)) = f∞ (µ(∞) 1 ∞ ) + ∂ f .µ(∞); µ(n) − µ(∞)Σ . Similarly, and using Lemma 4.3.5 Yn(µ(n)) = Yn(µ(∞)) + Op .∂Yn .µ(∞); µ(n) − µ(∞)ΣΣ = Yn(µ(∞)) + Op .ǁµ(n) − µ(∞)ǁL2 Σ . From the definition of Yn we also have fn(µ (n) ) = f∞(µ (n) ) + √n Yn(µ (n)). Substituting into the above we obtain fn(µ (n) ) = f∞(µ (∞) ) + ∂ . f∞(µ (∞) ; µ(n) − µ(∞) 1 . ) + √n Yn(µ (∞)) + Op √nǁµ (n) − µ(∞) ǁL2 Σ = fn(µ (∞) 1 2 ) + ∂ 2 f∞(µ (∞) ; µ(n) − µ(∞) ) + Op 1 √nǁµ (n) − µ(∞) ǁL2 Σ . Rearranging for ∂2f∞ and using fn(µ(n)) ≤ fn(µ(∞)) we have ∂ f∞(µ (∞) ; µ(n) − µ(∞) ) = 2 .fn(µ (n) ) − fn(µ (∞))Σ 1 + Op( √nǁµ (n) − µ(∞) ǁL2 ) ≤ Op( √nǁµ (n) − µ(∞) ǁL2 ). Recall that by Lemma 4.3.4 we have that ∂2f∞ is positive definite at µ(∞). Therefore:
Proof of Theorem 4. 2. The works of Xxx ([80],[81]) and Xxxxxxx [2] show that ∆d(n) ≥ 0 when d ≥ 31 and can be easily modified to show that the inequality is strict when n ≥ d + 6. For each remaining 4 ≤ d ≤ 30, we use Theorems 4.3 and 4.6 to compute the smallest n such that our bounds imply ∆d(n) > 0. We denote this n by Ω(d), and a C++ program computed the values of ∆d(n) ≤ Ωd(n), which then confirmed the remaining cases of the Xxxxx-Xxxxxxx Conjecture. As an example, we find that when d = 30, Ω(30) ≤ 9.77·106. To get this, we take δ = 10−10 and s1 = 5·10−11 in Theorem 4.3 and, in Theorem 4.6, s = .16906, s2 = .499999, ξ = .99, c = .375000001, and ν = 1. Other d are similar, and all satisfy Ω(d) ≤ Ω(30).
Proof of Theorem 4. 1 In this section we prove the algorithmic quasirandom lemma. The proof is based on iteratively applying Algorithms 2.7 and 4.3 and is similar to Szemeredi’s original proof of the regularity lemma. As in Xxxxxxxxx’s original proof the index function plays a pivotal role. Therefore, we will first prove some properties of this function which we will need later. Let us start with generalizing the definition of the index-function from (indicator function of) hypergraphs to arbitrary functions.
Proof of Theorem 4
