Spatial Random Processes Introduction Literature - R. van der Hofstad: Random graphs and complex nerworks. www.win.tue.nl/~hofstad/NotesRGCN2009.pdf B.Bollobas: Random graphs For probability background โ R.Durrett, probability theory and examples (2005) Main subject We will look at the Erdos-Renyi random graph (1960) ๐ Consider n vertices, let each one of the ( ) possible edges appear independently with 2 possibility ๐ โ [0,1] Such a graph is referred to as ๐บ(๐, ๐). Question: What is the size of the largest component? Denote it as ๐ถ๐๐๐ฅ of ๐บ(๐, ๐). ๐ ๐ We define: ๐ = , ๐ > 0 Exercise: Prove that ๐ธ[๐๐๐๐๐๐(๐ฃ)] = ๐โ1 ๐ ๐ We will see that: ๐โโ If ๐ < 1, than โ๐ถ๐ > 0 ๐ . ๐ก. โ๐ > 0. ๐[(๐ถ๐ โ ๐) log ๐ โค |๐ถ๐๐๐ฅ | โค (๐ถ๐ + ๐) log ๐] โ ๐โโ If ๐ > 1, then โ๐ถโฒ๐ > 0 ๐ . ๐ก. โ๐ > 0 ๐[(๐ โฒ๐ โ ๐)๐] โค |๐ถ๐๐๐ฅ | โค (๐ถ โฒ๐ + ๐)๐] โ 1 1 Fix vertex v. Do breadth first exploration of component ๐ถ(๐ฃ) of ๐บ(๐, ๐) containing v vertices. Can be either N (neutral), A(Active) or I(Inactive). At time ๐ก = 0, ๐ฃ โ ๐ด and all other vertices are in N. At time ๐ก โฅ 1, if ๐ด = ๐, terminate. Otherwise, choose ๐ค โ ๐ด. Put all N neighbors of ๐ in ๐ด. Put ๐ in ๐ผ. Note: Algorithm terminates after |๐ถ(๐ฃ)| steps. Suppose at time ๐ก, |๐| = ๐ โ ๐ vertices are left and |๐ด| โฅ 1. The number of vertices that go from N to A in stem ๐ก has the distribution: ๐ ๐ ๐ต๐๐(๐ โ ๐, ๐) = ๐ต๐๐ (๐ โ ๐, ) โ ๐ต๐๐(๐, ) ๐ ๐ ๐ Recall that ๐~๐ต(๐, ๐), ๐ โ [0,1] โ ๐[๐ = ๐] = ( ) ๐๐ (1 โ ๐)๐โ๐ , 0 โค ๐ โค ๐ ๐ Expect similarity with a branching process! Branching Process t=0 t=1 ๐ ๐ต๐๐(๐, ) ๐ ๐ ๐ต๐๐(๐, ) ๐ t=2 ๐๐๐(๐) ๐~๐๐๐(๐), ๐ โ (0, โ)๐๐ ๐[๐ = ๐] = ๐๐ ----โ๐ ๐ , ๐โโค ๐! ๐โโ ๐๐ ๐ โ๐ , ๐! ๐ Exercise: If ๐~๐ต๐๐(๐, ๐), then ๐[๐ = ๐] โ ๐โโค We will see that: If ๐ < 1, then branching process (๐๐๐ (๐)) dies out almost surely. If ๐ > 1, then it lives forever with probability >0. Further Questions 2 - ๐ = 1 โ |๐ถ๐๐๐ฅ | โ ๐3 - If ๐ = 1 + ๐ฟ๐ , ๐ฟ๐ โ 0 Other properties of ๐บ(๐, ๐) Add geometry to the graph Fix a large graph ๐บ๐ Retain (delete) every edge independently with probability ๐. If ๐บ๐ = ๐๐ (the full graph), we get ๐บ(๐, 1 โ ๐) Consider a random walk on ๐บ๐ , study the components ๐บ๐ \ {๐๐ก1 , โฆ , ๐๐ก๐ }, ๐ก๐ โฅ 0 Study random graphs s.t. ๐[๐๐๐๐๐๐(๐ฃ) โฅ ๐] โ ๐ โ๐ , ๐ > 0 - ๐โโ Branching Processes Time ๐ = 0, 1, โฆ ๐1,1 t=0 ๐2,1 ๐2,๐1,1 t=1 t=2 In generation ๐ โฅ 0, individual ๐ gives birth to ๐๐+1,๐ children, then dies. Formally, we define {๐๐,๐ |๐ โฅ 1, ๐ โฅ 1} as independent, identically distributed (iid) random variables with values in โค > 0 Define ๐๐ , (๐ โฅ 0) recursively by ๐0 = 1 โฎ ๐๐=โ๐๐ โ1 ๐ ๐=1 ๐,๐ ๐0 = to the total number of individuals in generation 1. Notation ๐ = ๐1,1 (since all distribute identically) ๐๐ = ๐[๐ = ๐] =probability that individual has ๐ children. ๐ = ๐ [โ{๐๐ = 0}] = ๐[๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐๐ข๐ก] ๐โฅ1 For ๐ โ [0,1] ๐บ๐ฅ (๐ ) = ๐ธ[๐ ๐ฅ ] โoffspring generation functionโ Theorem 1.1: - If ๐ธ[๐ฅ] < 1, ๐กโ๐๐ ๐ = 1 - If ๐ธ[๐ฅ] > 1, ๐กโ๐๐ ๐ < 1 - If ๐ธ[๐ฅ] = 1, ๐๐๐ ๐[๐ = 1] < 1, ๐กโ๐๐ ๐ = 1 Moreover, ๐ is the smallest solution in [0,1] of ๐ = ๐บ๐ฅ (๐) Exercise 1.1: Prove ๐ธ[๐๐ ] = ๐๐ , where ๐ = ๐ธ[๐] Exercise 1.2: For ๐ < 1, prove ๐ธ[๐] = --------End of lesson 1 1 1โ๐ where ๐ = โโ ๐=0 ๐๐ Reminder - {๐ฅ๐,๐ }๐,๐โฅ1 ๐๐๐, โค โฅ 0 โ ๐ฃ๐๐๐ข๐๐ ๐๐โ1 - ๐0 = 1, ๐๐ = โ๐=1 ๐๐,๐ - ๐ = ๐1,1 - ๐๐ = ๐[๐ = ๐], ๐ = ๐๐ โค โฅ 0 - ๐ = ๐[โ๐โฅ0{๐๐ = 0}] - ๐บ๐ฅ (๐ ) = ๐ธ[๐ ๐ฅ ], ๐ โ [0,1] Theorem 1.1 (survival vs. extinction): if ๐ธ[๐] < 1 then ๐ = 1, If ๐ธ[๐] > 1 then ๐ < 1, If ๐ธ[๐] = 1 and ๐[๐ = 1] < 1 then ๐ = 1 ๐ is the smallest solution in [0,1] of ๐ = ๐บ๐ฅ (๐) Example: ๐~๐๐๐(๐), ๐ > 0 โ ๐ ๐บ๐ฅ (๐ ) = โโ ๐=0 ๐ ๐๐ = โ๐=0 ๐ ๐ ๐๐ โ๐ ๐ ๐! = ๐ ๐(๐ โ1) Proof: First prove that ๐ = ๐บ๐ฅ (๐) Set ๐๐ = ๐ [ ๐๐ = 0 ] โ ๐๐๐๐๐๐๐ ๐๐๐ ๐๐ ๐ By ๐ additivity, ๐๐ โ ๐ as ๐ โ โ. Let ๐บ๐ (๐ ) = ๐ธ[๐ ๐๐ ], ๐ โ [0,1] (โ means that ๐๐ โค ๐๐+1 , ๐๐ โ ๐) Note that is we set ๐บ๐ (0) = ๐๐ ๐ ๐บ๐ (0) = โโ ๐=1 0 ๐[๐๐ = ๐] = ๐๐ + 0 + 0 + โฏ + 0 = ๐๐ . ๐๐ ๐บ๐ (๐ ) = โโ ๐=0 ๐ธ[๐ 1{๐๐ =๐} ] , where 1๐ด (๐ค) = { TODO: Draw the full drawing ๐ก=1 ๐ก=๐ 1 ๐๐ ๐ค โ ๐ด 0 ๐๐ ๐ค โ ฮฉ\๐ด (๐1 ) (1) ๐๐ = ๐๐ + โฏ + ๐๐ (๐) where ๐๐ Is the number of generation n-individuals descending from (๐) (๐) generation 1-individual ๐ and ๐๐ = ๐๐โ1 (๐) (๐๐ ) ๐โฅ0 ๐ ๐ ๐๐โ1 ] ๐๐โ1 ] are independent of ๐1 hence ๐บ๐ (๐ ) = โโ ๐๐ = โโ ๐๐ = ๐=0 ๐ธ[๐ ๐=0 ๐ธ[๐ ๐ โโ ๐=0 ๐บ๐โ1 (๐ ) ๐๐ = ๐บ๐ฅ (๐บ๐โ1 (๐ )). = ๐บ1 (๐บ๐โ1 (๐ )) ๐ = 0: ๐๐ = ๐บ1 (๐๐โ1 ), ๐ โ โ ๐ โ ๐บ1 (๐) By the dominated convergence theorem. ๐บ1 (๐๐โ1 ) = ๐ธ[(๐๐โ1 )๐1 ] ๐โโ (๐๐โ1 ) ๐ง1 โ ๐ ๐1 Let ๐ โ [0,1], ๐ = ๐บ๐ฅ (๐) Now we need to show that ๐ โค ๐, or equivalently, ๐๐ โค ๐ โ๐. Indeed: ๐0 = 0 โค ๐ โ [0,1], if ๐๐โ1 โค ๐, then ๐๐ = ๐บ๐ฅ (๐๐โ1 ) โค ๐บ๐ฅ (๐) ๐๐ฆ ๐๐ ๐ ๐ข๐๐๐ก๐๐๐ = ๐. Case ๐[๐ โ {0,1}] = 1 ๐[๐ = 0] = ๐ > 0 ๐โโ ๐๐ = 1 โ ๐[๐๐ > 0] = 1 โ (1 โ ๐)๐ โ 1. Now assume ๐[๐ฅ โค 1] < 1. ๐บ๐ฅโฒ (๐ ) = ๐ธ[๐๐ ๐โ1 ] > 0 ๐บ๐ฅโฒโฒ (๐ ) = ๐ธ[๐(๐ โ 1)๐ ๐โ2 ] > 0, ๐ โ (0,1). (Ex: Prove this by using the mean value theorem and the dominant convergence theorem.) ๐บ๐ฅ (1) = 1 TODO: Draw the graph of the function ๐บ๐ฅ (๐ ) Either ๐ = ๐บ๐ฅ (๐ ) has one or two solutions in [0,1]. One solution โ ๐บ๐ฅโฒ (1) โค 1 ๐บ๐ฅโฒ (1) = ๐ธ[๐] Exercise 1.4: Compute ๐ If ๐0 = 1 โ ๐, ๐2 = ๐, ๐ โ [0,1] โBinary branchingโ. Exercise 1.5: Prove ๐[๐๐ > 0] โค ๐๐ , ๐ = ๐ธ[๐]. Hint: use the Chebishev inequality and the previous exercise. Dominated Convergence Theorem ๐โโ Let (๐๐ )๐โฅ1 be a sequence of random variables, ๐๐ โ ๐ a.s. Assume that โ a random variable ๐ โฅ 0 almost surely ๐ธ[๐] < โ such that |๐๐ | โค ๐ โ๐ almost ๐โโ surely, Then ๐ธ[๐๐ ] โ ๐ธ[๐]. Exercise 1.6: Use the Dominated Convergence Theorem to fill in the details of the proof of theorem 1.1. Change of Notation (random walk perspective) TODO: Draw the drawing ๐ก=0 ๐ก =n Redenote ๐๐,๐ as they appear in breadth-first search exploration of the tree. ๐. ๐. (๐1,1 , ๐2,1 , โฆ , ๐2,2 , ๐3,1 , โฆ ) =: (๐1 , ๐2 , ๐3 , โฆ ) Define the random variables ๐๐ , ๐ผ๐ , ๐ โฅ 0 such that ๐๐ = ๐(๐๐ ,๐ผ๐) Observation: (๐๐ , ๐ผ๐ ) depend only on (๐1 , โฆ , ๐๐โ1 ) Lemma 1.7: (๐๐ )๐โฅ1 are iid with the same distribution as ๐. Proof: Let ๐1 , โฆ , ๐๐ โ โค โฅ 0 ๐[๐1 = ๐ฅ1, โฆ , ๐๐ = ๐ฅ๐ ] = โ โ ๐ [(๐1 = ๐ฅ1 , โฆ , ๐๐โ1 = ๐ฅ๐โ1 , ๐๐ = ๐, ๐ผ๐ = ๐ ) ๐๐,๐ = ๐ฅ๐ ] ๐โฅ1 ๐โฅ1 ๐โ1 โ ๐ด ๐โ1 ๐ด depends only on ๐ ({๐๐,๐ }๐=1,๐=1 {๐๐,๐ }๐=1 ) โ๐ โ๐ ๐[๐ด]๐[๐ = ๐ฅ๐ ] = ๐[๐1 = ๐ฅ1 , โฆ , ๐๐โ1 = ๐ฅ๐โ1 ]๐[๐ = ๐ฅ๐ ] ๐[๐ = ๐ฅ๐ ] ๐๐๐๐ข๐๐ก๐๐๐ = ๐[๐ = ๐ฅ1 ] โ โฆ โ Independence of ๐๐,๐ Notation (ctd): (๐๐ )๐โฅ0 equals to the number of active individuals at stage ๐. ๐0 = 1 ๐๐ = ๐๐โ1 + (๐๐ โ 1), ๐ โฅ 1 = (๐1 , โฆ , ๐๐ ) โ (๐ โ 1) = number of active individuals. ๐ = (โ๐โฅ0 ๐๐ ) = inf{๐ โฅ 0 โถ ๐๐ = 0} ๐0 = 1 ๐1 = 1 + (2 โ 1) ๐2 = 2 + (2 โ 1) ๐3 = 3 + (0 โ 1) ๐4 = 2 + (0 โ 1) = 1 ๐ 5 = 1 + (0 โ 1) = 0 ------- end of lesson 2 In the proof of theorem 1.1 we showed: ๐บ๐ (๐ ) = ๐ธ[๐๐๐ ] = ๐บ๐ (๐บ๐โ1 (๐ )), ๐ โ [0,1] Exercise 1.7: Set ๐บ๐ (๐ ) = ๐ธ[๐ ๐ ], ๐ โ [0,1] Prove that ๐บ๐ (๐ ) = ๐ โ ๐บ๐ (๐บ๐ (๐ )), where ๐บ๐ (๐ ) = ๐ธ[๐ ๐ ] Exercise 1.8: Prove the following : If ๐ = ๐ธ[๐] = 1, then ๐ธ[๐] = โ (hint: Prove first that ๐ธ[๐] = 1 + ๐ธ[๐]) Exercize 1.9: Recall ๐ธ[๐๐ ] = ๐๐ - ๐ Prove that (๐๐ = ๐๐๐ ) ๐โฅ0 is a martingale (w.r.t. natural filtration). Show that: lim ๐๐ = ๐โ exists and 0 โค ๐๐ โค โ almost surely. ๐โโ - For ๐ > 1and ๐ธ[๐ 2 ] < โ show that (๐๐ )๐โฅ0 is bounded in ๐ฟ2 . Use theorem 1.1 and onvergence theorem (๐ฟ2 ) to prove that ๐[๐โ = 0] = ๐ = ๐[๐ < โ]. Deduce that {๐โ = 0} = {๐ < โ} almost surely. (โ ๐๐ ~(๐โ โ ๐๐ )๐๐ {๐=โ} ) Theorem 1.10: (Extinction with large total progeny) ๐ โ๐ผ๐ If ๐ = ๐ธ[๐] > 1, then ๐[๐ โค ๐ < โ] โค 1โ๐ ๐ผ where ๐ผ = sup(๐ก โ log ๐ธ[๐ ๐ก๐ ]) > 0 ๐ Remark: In ๐บ (๐, ๐) , ๐ > 1, we will see that โ๐ > 0 ๐ . ๐ก. there are no components of size 1 between ๐ log ๐ and ๐ โ ๐, with high probability as ๐ โ โ. Lemma 1.11: (Cramer/Chernoff bound): Let {๐๐ }โ ๐=1 be iid. 1. For any ๐ โ (๐ธ[๐1 ], โ), ๐[โ๐๐=1 ๐๐ โฅ ๐๐] โค ๐ โ๐๐ผ(๐) 2. For any ๐ โ (โโ, ๐ธ[๐1 ]), ๐[โ๐๐=1 ๐๐ โค ๐๐] โค ๐ โ๐๐ผ(๐) Where ๐ผ(๐) = sup(๐ก๐ โ log ๐ธ[๐ ๐ก๐1 ]) in (1) tโฅ0 and ๐ผ(๐) = sup(๐ก๐ โ log ๐ธ[๐ ๐ก๐1 ]) in (2) tโค0 Proof of Lemma 1.11: ๐ ๐ ๐ ๐ [โ ๐๐ โฅ ๐๐] = ๐ [exp (โ๐ก๐๐ + ๐ก โ ๐๐ ) โฅ 1] = ๐ธ[1๐ด ] โค ๐ธ 1๐ด โ exp (โ๐ก๐๐ + ๐ก โ ๐๐ ) โ ๐=1 ๐=1 ๐=1 โ๐กโฅ0 [ ] โฅ1 ๐ ๐ โค ๐ธ exp (โ๐ก๐๐ + ๐ก โ ๐๐ ) = ๐ โ ๐=1 [ ] โฅ0 ๐ก๐๐ ]) = exp(โ๐(๐ก๐ โ log ๐ธ[๐ ) Now optimize over ๐ก โฅ 0 โ(1). โ๐ก๐๐ ๐ธ [โ ๐ ๐ ๐ก๐๐ ]=๐ โ๐ก๐๐ ๐=1 โ ๐ธ[๐ ๐ก๐๐ ] ๐=1 (2) is left as an exercise! Proof of theorem 1.10: โ โ โ ๐[๐ โค ๐ < โ] = โ ๐ [ ๐ โ= ๐ ] โค โ ๐[๐1 + โฏ + ๐๐ = ๐ โ 1] = โ ๐[๐1 + โฏ + ๐๐ โค ๐] ๐=๐ โ{๐๐ =0} ๐=๐ ๐=๐ ๐๐ = (๐1 + โฏ + ๐๐ ) โ (๐ โ 1) ๐๐๐๐๐ 1.11,(2),๐=1 ๐[๐ โค ๐ < โ] โค โ โ ๐ โ๐๐ผ(1) ๐=๐ Define โ ๐(๐ก) = ๐ก โ log ๐ธ[๐ ๐ก๐ ] , ๐ก โค 0 ๐(0) = 0 โ log 1 = 0 1 ๐ โฒ (๐ก) | = 1 โ โ ๐ธ[๐๐ 0๐ ] = 1 โ ๐ธ[๐] < 0 0๐ ] ๐ธ[๐ ๐ก=0 Since the derivative is negative, there is some negative value where ๐(๐ก) approaches from above. So โ๐ก < 0: ๐(๐ก) > 0 โ ๐ผ(1) > 0 Duality Principle Assume ๐ โ (0,1) (๐ = ๐[๐ < โ]) Condition the branching process to die out. (only consider the probability ๐[๐ด|๐ < โ] = ๐[๐ดโฉ{๐<โ}] ๐[๐<โ] (A event) Question: How does this change the branching process? Definition 1.12: (Conjugate Distribution) Let ๐๐ฅ = ๐[๐ = ๐ฅ], ๐ฅ โ โคโฅ0 and assume ๐0 > 0 (โ ๐ > 0). Then the conjugate distribution (๐๐ฅโฒ )๐ฅโโคโฅ0 associated to ๐ is defined as ๐๐ฅโฒ = ๐ ๐ฅโ1 ๐๐ฅ , ๐ฅ โ โคโฅ0 Lemma 1.13: โ ๐๐ฅโฒ = 1 ๐ฅโฅ0 Proof: โ ๐๐ฅโฒ = โ ๐ ๐ฅโ1 ๐๐ฅ = ๐ฅโฅ0 ๐ฅโฅ0 1 1 1 โ ๐ ๐ฅ ๐๐ฅ = ๐บ๐ (๐) = โ ๐ = 1 ๐ ๐ ๐ ๐ฅโฅ0 Denote the branching process with offspring distribution ๐โฒ (๐โฒ branching process) by โฒ {๐๐,๐ }๐โฅ1,๐โฅ1 Lemma 1.14 (๐ > 0) ๐[๐ โฒ < โ] = 1 Proof: Let ๐ โ [0,1], assume that ๐บ๐ โฒ (๐ ) = ๐ . Want to show โ ๐ = 1, use theorem 1.1. โ โ 1 ๐บ๐ โฒ (๐ ) = โ ๐ ๐[๐ = ๐] = โ ๐ ๐ (๐ ๐โ1 ๐๐ ) = ๐บ๐ (๐ ๐) = ๐ ๐ ๐ โฒ ๐=0 ๐=0 So ๐ ๐โฅ๐ ๐บ๐ (๐ ๐) = ๐ ๐ โ โ ------ End of lesson 3 ๐ ๐ โฅ ๐ โ ๐ = 1 Duality Principle Assume that ๐๐[๐ < โ] โ (0,1) Given ๐๐ฅ = ๐[๐ = ๐ฅ], ๐ฅ โฅ 0, ๐0 > 0 Define conjugate distribution ๐โฒ by ๐๐ฅโฒ = ๐ ๐ฅโ1 ๐๐ฅ , ๐ฅ โ โคโฅ0 โฒ {๐๐,๐ }๐โฅ1,๐โฅ1 ๐โฒ - A branching process (๐ โฒ , ๐๐โฒ , ๐1โฒ , ๐2โฒ โฆ ) then ๐[๐ โฒ < โ] = 1 Theorem 1.15: (duality princinple) Let ๐ = (๐๐ฅ )๐ฅโโคโฅ0 be a distribution on โคโฅ0 with ๐0 > 0 and let ๐โฒ be its conjugate distribution. Let (๐๐ )๐โฅ1 be a branching process. Let (๐๐โฒ )๐โฅ1 be a branching process Then ๐[(๐1 , โฆ , ๐ ๐ โฒ ) = (๐ฅ1 , โฆ , ๐ฅ๐ก )|๐ < โ] = ๐[(๐1โฒ , โฆ , ๐๐โฒ ) = (๐ฅ1 , โฆ , ๐ฅ๐ก )|๐ < โ] For ๐ฅ1 , โฆ , ๐ฅ๐ โ โคโฅ0 . In words, ๐-branching process conditioned to die out is distributed as a ๐โฒ branching process. Proof: Denote ๐ป = (๐1 , โฆ , ๐๐ ), ๐ป โฒ = (๐1โฒ , โฆ , ๐๐โฒ โฒ ), โ = (๐ฅ1 , โฆ , ๐ฅ๐ก ) Recall: ๐๐ = (๐1 + โฏ + ๐๐ ) โ (๐ โ 1) ๐ = inf{๐ โฅ 1|๐๐ = 0} ๐ ๐ = (๐ฅ1 , โฆ , ๐ฅ๐ ) โ (๐ โ 1) Assume that ๐๐ > 0 for 1 โค ๐ < ๐ก and ๐๐ก = 0. Otherwise (1) otherwise the equation is true trivially (0 = 0) ๐ก ๐ก ๐=1 ๐=1 ๐[๐ป = โ, ๐ < โ] ๐ต๐ฆ ๐๐๐๐๐ 1.7 1 1 ๐[๐ป = โ|๐ < โ] = = โ ๐๐ฅ๐ = โ ๐๐ฅโฒ ๐ ๐1โ๐ฅ๐ ๐[๐ < โ] ๐ ๐ ๐ก = 1 ๐กโโ๐ก ๐ฅ โ ๐ ๐=1 ๐ โ ๐๐ฅโฒ ๐ ๐ โ ๐=1 ๐[๐ป โฒ =โ] โ๐ก๐=1 ๐ฅ๐ But ๐ก โ = ๐ก โ (๐ ๐ก + (๐ก โ 1)) = ๐ก โ ๐ก + 1 = 1 โฒ So it equals: ๐[๐ป = โ] Which is what we needed to prove. Exercise 1.16: Let ๐ be ๐๐๐(๐) - distributed, ๐ > 0. Recall from Lemma 1.11 ๐ผ(๐) = sup(๐ก๐ โ log ๐ธ[๐ ๐ก๐ฅ ]) , ๐ > 0 (๐ก โฅ 0 if ๐ > ๐) (๐ก โค 0 if ๐ < ๐) Show that in both cases (๐ > ๐, ๐ < ๐) ๐ ๐ผ(๐) = sup (๐ก๐ โ log ๐ธ[๐ ๐ก๐ฅ ]) = ๐ โ ๐ + ๐ log ( ) ๐ tโRโ Exercise 1.17 (coupling) Let (๐๐ฅ )๐ฅโโค , (๐๐ฅ )๐ฅโโค be different probability distributions on โค. We want a random variable (๐, ๐), โค2 valued, such that ๐[๐ = ๐ฅ] = ๐๐ฅ , ๐ฅ โ โคโฅ0 ๐[๐ = ๐ฅ] = ๐๐ฅ , ๐ฅ โ โคโฅ0 ๐[๐ โ ๐] as small as possible. Prove that: 1 2 (๐๐ฅ โmin{๐๐ฅ ,๐๐ฅ })(๐๐ฅ โmin{๐๐ฅ ,๐๐ฅ }) ๐๐๐ (๐,๐) (i) โ๐ฅโโค(๐๐ฅ โ min{๐๐ฅ , ๐๐ฅ }) = โ๐ฅโโค(๐๐ฅ โ min{๐๐ฅ , ๐๐ฅ }) = โ๐ฅโโค|๐๐ฅ โ ๐๐ฅ | = ๐๐๐ (๐, ๐) (ii) ๐๐ฅ,๐ฆ = min{๐๐ฅ , ๐๐ฅ } if ๐ฅ = ๐ฆ, and (iii) otherwise. Is a distribution on โค2 With ๐[(๐, ๐) = (๐ฅ, ๐ฆ)] = ๐๐ฅ,๐ฆ , (๐ฅ, ๐ฆ) โ โค2 get ๐[๐ โ ๐] = ๐๐๐ (๐, ๐) and this is the best. if ๐[(๐, ๐) = (๐ฅ, ๐ฆ)] = ๐โฒ๐ฅ,๐ฆ , ๐[๐ = ๐ฅ] = ๐๐ฅ , ๐[๐ = ๐ฅ] = ๐๐ฅ โ ๐[๐ โ ๐] โฅ ๐๐๐ (๐, ๐) Possible for any random variable (๐ โฒ , ๐โฒ) with values in โค2 , with ๐[๐ โฒ = ๐ฅ] = ๐๐ฅ , ๐[๐ โฒ = ๐ฅ] = ๐๐ฅ โ ๐[๐ โฒ โ ๐ โฒ ] > ๐๐๐ (๐, ๐) Poisson branching processes Notation: Use superscript โ*โ for Poisson branching processes. (๐๐โ , ๐๐โ , ๐ โ , โฆ ) Recall: ๐๐ โ๐ ๐ , ๐! - ๐[๐ โ = ๐] = - ๐ธ[๐ โ ] = ๐ ๐บ๐ โ (๐ ) = ๐ ๐(๐ โ1) , ๐ โฅ 0 ๐ โ โคโฅ0 Definition 2.1: Call 0 < ๐ < 1 < ๐ a conjugate pair if ๐๐ โ๐ = ๐๐ โ๐ Note: ๐: ๐ฅ โ ๐ฅ๐ โ๐ฅ is strictly increasing on [0,1] It is strictly decreasing on [1, โ] ๐(0) = 0, ๐(โ) = 0 TODO: Draw functionโฆ So for any ๐ > 1, there is a unique conjugate ๐ โ [0,1) and vice-versa. Theorem 2.2: (Poisson duality principle): Let ๐ < 1 < ๐ be a conjugate pair The ๐๐๐(๐) - a branching process conditioned on extinction is distributed as ๐๐๐(๐) - a branching process. Proof: By theorem 1.15, The conditioned process is a branching process with offspring distribution ๐๐ฅโฒ = ๐๐๐ฅโ1 ๐๐ฅ โ๐ ๐ , ๐ฅ! ๐ฅ > 0 where ๐๐ = extinction probability for ๐๐๐(๐)- branching process. ๐๐ โ (0,1) i.e. (๐๐๐ )๐ฅ โ๐ 1 ๐๐ฅโฒ = ๐ ( ) ๐ฅ! ๐๐ By theorem 1.1: ๐๐ = ๐บ๐ โ (๐๐ ) = ๐ ๐(๐๐ โ1) So ๐๐ฅโฒ = (๐๐๐ )๐ฅ ๐ฅ! ๐ โ๐ โ 1 ๐ ๐(๐๐ โ1) = (๐๐๐ )๐ฅ ๐ฅ! ๐ โ๐๐๐ , ๐ฅ โฅ 0 Hence, ๐โฒ = ๐๐๐(๐๐๐ ) ? ๐๐๐ ๐ โ๐๐๐ = ๐๐ โ๐ Have to prove ๐๐๐ = ๐ ๐๐๐ ๐ โ๐๐๐ = ๐๐ ๐(๐๐ โ1) ๐ โ๐๐๐ = ๐๐ โ๐ So, ๐๐๐ โ ๐ (since ๐ > ๐ (theorem 1.1) โ ๐ธ[๐ โ ] > 1 โ ๐๐ > 1) And (๐๐๐ )๐ โ(๐๐๐ ) = ๐๐ โ๐ Since ๐ฅ๐ โ๐ฅ โ on [0,1], โ on [1, โ] There is at most 1 ๐ฅ diferent from ๐ such that ๐ฅ๐ โ๐ฅ = ๐๐ โ๐ = ๐ Hence, ๐๐๐ = ๐ โ Theorem (Cayleyโs formula): The number of labeled trees on ๐ vertices is equal to ๐๐โ2 TODO: Draw a tree Theorem 3.15: In reference ---- End of lesson 4 โ โ ๐ = โ ๐๐โ = inf{๐ โฅ 0: (๐1 + โฏ + ๐๐ ) โ (๐ โ 1) = 0} ๐=1 Theorem 2.3 (Total Progeny of Poisson Branching Process): For ๐๐๐(๐) - branching process. ๐ > 0 (๐๐ )๐โ1 โ๐๐ ๐[๐ โ = ๐] = ๐ , ๐! ๐โฅ1 Proof: By Induction on ๐. โ Check for ๐ = 1: ๐[๐ โ = 1] = ๐[๐1,1 = 0] = ๐ โ๐ which fully agrees with the formula. ๐ โฅ 2, assume the result holds for all ๐โฒ < ๐. ๐โ1 โ ๐[๐ = ๐] = โ ๐[๐ โ = ๐, ๐1,1 = ๐] โ ๐=1 If we observe the children that โsprungโ from all children of the first one, the sum of all vertices is their sum + 1 (for the root) equals the sum of all vertices in the graph. โ ๐ โ is distributed as 1 + ๐ (1) + โฏ + ๐ (๐1,1 ) , where {๐ (2) }๐โฅ1 are iid with the same distribution as ๐โ ๐โ1 โ ๐[๐ = ๐] = โ ๐[๐ (1) + โฏ + ๐ (๐) = ๐ โ 1, ๐1,1 = ๐] โ ๐=1 But these random variables are independent (by our assumption) ๐โ1 =โ โ ๐[๐ (1) = ๐1 , โฆ , ๐ (๐) = ๐๐ , ๐1,1 = ๐] = โ ๐=1 (๐1 ,โฆ,๐๐ ) ๐๐ โฅ1, โ๐ ๐๐ =๐โ1 ๐โ1 โ โ โ ๐[๐ (1) = ๐1 ] โ โฆ โ ๐[๐ (๐) = ๐๐ ] โ ๐[๐1,1 = ๐] ๐=1 (๐1 ,โฆ,๐๐ ) ๐๐ โฅ1, โ๐ ๐๐ =๐โ1 Now we can use our induction hypothesis! Each of the ๐๐ โs is smaller than ๐. ๐โ1 โ ๐ โ ๐โ1โ๐ โ๐(๐โ1) ๐ ๐ โ โ( ๐=1 (๐1 ,โฆ,๐๐ ) ๐๐ โฅ1, โ๐ ๐๐ =๐โ1 So, so far we have shown that: ๐=1 (๐๐ )๐๐โ1 ๐๐ โ๐ )โ ๐ ๐๐ ! ๐! ๐โ1 โ ๐โ1 โ๐๐ ๐[๐ = ๐] = ๐ ๐ 1 โ ๐! ๐=1 โ ๐ โ โ( (๐1 ,โฆ,๐๐ ) ๐=1 ๐๐ โฅ1 โ๐ ๐๐ =๐โ1 (๐๐ )๐๐ โ1 ) = ๐๐โ1 ๐ โ๐๐ โ ๐๐ ๐๐ ! =:๐๐ Lemma 2.4: Let ๐ฟ๐ be the number of labeled trees on ๐ vertices. Then, ๐ฟ๐ = ๐๐โ2 = (๐ โ 1)! ๐๐ ๐๐โ2 With the lemma 2.4, we get ๐[๐ โ = ๐] = ๐๐โ1 ๐ โ๐๐ โ (๐โ1)! = (๐๐)๐โ1 ๐! ๐ โ๐๐ โ (theorem 2.3). How do we prove lemma 2.4? We use lemma 2.5. Lemma 2.5 (Cayleyโs Formula): ๐ฟ๐ = ๐๐โ2 Example: ๐=5 TODO: Draw graph Proof that lemma 2.5 โ Lemma 2.4: For ๐ โฅ 1, ๐1 , โฆ , ๐๐ , ๐๐ โฅ 1, โ๐ ๐๐ = ๐ โ 1, define ๐ก(๐1 ,โฆ,๐๐) =number of labeled trees of ๐ vertices such that ๐ฃ1 has ๐ neighbors, (tree\{๐ฃ1 }) โ has components with ๐1 , โฆ , ๐๐ vertices. To choose such a tree: 1. Split {๐ฃ2 , โฆ , ๐ฃ๐ } into sets of size ๐1 , โฆ , ๐๐ 2. Choose ๐ labeled trees of size ๐1 , โฆ , ๐๐ 3. Choose a vertex in each of the ๐ trees, and connect it to ๐ฃ1 ๐ ๐ก(๐1 ,โฆ,๐๐) (๐๐ )๐๐โ1 ๐โ1 ๐ โ2 ๐ โ2 =( ) โ (๐1 1 โ โฆ โ ๐๐ ๐ ) โ (๐1 โ โฆ โ ๐๐ ) = (๐ โ 1)! โ ( ) ๐1 , โฆ , ๐๐ ๐๐ ! ๐=1 Hence, ๐โ1 ๐ฟ๐ = โ ๐โ1 โ ๐๐ โฅ1 ๐=1 โ๐ ๐๐ =๐โ1 (๐๐๐๐๐๐๐๐๐!) Now weโre done! โ ๐ฟ๐ = (๐ โ 1)! โ ๐๐ ๐ก(๐1 ,โฆ,๐๐) = โ ๐=1 1 ๐! โ ๐๐ โฅ1 โ๐ ๐๐ =๐โ1 ๐๐๐๐๐๐๐! ๐ก(๐1 ,โฆ,๐๐) Lemma 2.5 (๐ฟ๐ = ๐๐โ2 , ๐ โฅ 1): Definition: A rooted tree is a tree with one distinguished vertex called โthe rootโ and the edges are all oriented, and oriented away from the root. Example: TODO: Draw an oriented tree Itโs not possible that a vertex will have 2 incoming edges. So a vertex has 1 incoming degree. For labeled vertices ๐ฃ1 , โฆ , ๐ฃ๐ , ๐ธ = {โฉ๐ฃ๐ , ๐ฃ๐ โช, ๐ โ ๐} ๐ ๐ = |{(๐1 , โฆ , ๐๐ ) โ ๐ธ ๐โ1 |({๐ฃ1 , โฆ , ๐ฃ๐ }, {๐1 , โฆ , ๐๐โ1 }) โ ๐๐ ๐ ๐๐๐๐ก๐๐ ๐ก๐๐๐}| Example: TODO: Draw vertices example To choose such a sequence as (was supposed to be drawn) above: 1. Choose a labeled tree on {๐ฃ1 , โฆ , ๐ฃ๐ } 2. Choose a root 3. Choose order in which to add the ๐ โ 1 edges. Hence, ๐ ๐ = ๐ฟ๐ โ ๐ โ (๐ โ 1)! = ๐ฟ๐ โ ๐! Alternatively, ๐ ๐ = ๐ ๐โฒ , where ๐๐โฒ is the number of rooted trees after you remove the last ๐ edges. ๐ ๐โฒ = |{(๐1 , โฆ , ๐๐ ) โ ๐ธ ๐โ1 |({๐ฃ1 , โฆ , ๐ฃ๐ }, {๐1 , โฆ , ๐๐โ1โ๐ }) โ ๐๐ ๐ ๐๐๐๐ก๐๐ ๐๐๐๐๐ ๐ก ๐ค๐๐กโ ๐ + 1 ๐๐๐๐๐๐๐๐๐ก๐ โ๐ = 0, โฆ , ๐ โ 1}| ๐ ๐ โค ๐ ๐โฒ obviously (every sequence in a set of ๐ ๐ can generate a sequence in ๐ ๐โฒ ) But also ๐ ๐โฒ โค ๐ ๐ , since for ๐ = 1, every forest is a rooted tree. Suppose edges ๐1 , โฆ , ๐๐โ1โ๐ have been added. Number of choices for ๐๐โ๐ = (๐ข, ๐ฃ) โ ๐ โ โ ๐ ๐ข ๐๐๐๐๐ก๐๐๐๐ฆ ๐ฃ ๐๐๐ฆ ๐๐๐๐ก ๐๐กโ๐๐ ๐กโ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐ถ(๐ข) Hence ๐โ1 ๐๐ = ๐ ๐โฒ = โ(๐๐) = ๐๐โ1 โ (๐ โ 1)! = ๐๐โ2 ๐! = ๐ฟ๐ (๐!) ๐=1 --- end of lesson 5 Theorem 2.3: For ๐๐๐(๐) = ๐ต. ๐. ๐[๐ โ = ๐] = (๐ > 0), (๐๐ )๐โ1 โ๐๐ ๐ , ๐! ๐โฅ1 Proof of Lemma 2.4: (Correction): ๐ โฅ 1, ๐1 , โฆ , ๐๐ , ๐๐ โฅ 1, โ ๐๐ = ๐ โ 1 ๐ TODO: Draw correction drawing ๐ก(๐1 , โฆ , ๐๐ ) = number of labeled trees on ๐ vertices {๐ฃ1 , โฆ , ๐ฃ๐ }, where ๐ฃ1 belongs to labeled edges 1, โฆ , ๐. And ๐ฃ1 has ๐๐ descendants attached to edge ๐, ๐ โ {1, โฆ , ๐} Reminder of what we did last time: ๐โ1 ( ) โ๐1 , โฆ , ๐๐ ๐ก(๐1 , โฆ , ๐๐ ) = ๐๐ ๐ค๐๐ฆ๐ ๐ก๐ ๐๐๐๐ก๐๐ก๐๐๐ {๐ฃ2 ,โฆ,๐ฃ๐ } ๐๐๐ก๐ ๐๐๐๐ข๐๐ ๐๐ ๐ ๐๐ง๐ ๐1 ,โฆ,๐๐ ๐1 โ2 ๐ โ2 = ๐ × โฆ × ๐๐ ๐ โ1 ๐๐ ๐โ๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐ ๐ ๐ข๐๐ก๐๐๐๐ × (๐1 × โฆ × ๐๐ ) โ ๐๐ ๐โ๐๐๐๐๐ ๐๐ ๐๐๐๐โ๐๐๐๐ ๐๐ ๐ฃ1 ๐ Remark: (๐ , โฆ , ๐ ) - is the number of ways to partition ๐ objects into ๐ labeled bins of sizes 1 ๐ ๐1 , โฆ , ๐๐ And the number of labeled trees on ๐ vertices equals ๐โ1 โ ๐=1 1 ๐! โ ๐ก(๐1 ,โฆ,๐๐) โ (๐1 ,โฆ,๐๐ ) โ๐๐โฅ1,โ๐ ๐โ1 # ๐๐ ๐ก๐๐๐๐ ๐ค๐๐กโ deg(๐ฃ1 )=๐ Corollary 2.6 (differentiability of the extinction probability): Let ๐(๐) = ๐๐ [๐ โ < โ], ๐ > 1 And let 0 < ๐ < 1 be the conjugate of ๐ (cf. definition 2.1) So ๐๐ โ๐ = ๐๐ โ๐ Then โ๐โฒ (๐) = ๐(๐)(๐ โ ๐) <โ ๐(1 โ ๐) Proof: Assume ๐ > 1 + ๐, ๐ > 0 By theorem 2.3: โ ๐(๐) = โ ๐=1 (๐๐)๐โ1 โ๐๐ ๐ ๐! By using the mean-value theorem, and the dominant convergence theorem, we can check that we can differentiate term-by-term. So: โ ๐ โฒ (๐) =โ ๐=1 (๐๐)๐โ1 โ๐๐ (๐ โ 1)๐๐โ2 ๐๐โ1 โ๐๐ ๐๐ โ ๐ = ๐! ๐! ๐๐ฆ ๐กโ๐๐๐๐๐ 1.15 (๐๐ข๐๐๐๐ก๐ฆ) 1 1 1 1 ๐ธ๐ [๐ 1{๐ โ <โ} ] (1 โ ) + ๐(๐) = ๐ธ๐ [๐ โ |๐ โ < โ] โ ๐(๐) (1 โ ) + ๐(๐) = ๐ ๐ ๐ ๐ 1 1 ๐(๐) ๐๐ฆ ๐ธ๐ฅ.1.3(?) ๐(๐) (1 โ ๐) ๐(๐) ๐(๐)(๐ โ 1) + ๐(๐)(1 โ ๐) โ ๐ธ๐ [๐ ] โ ๐(๐) (1 โ ) + = + = ๐ ๐ 1โ๐ ๐ ๐(1 โ ๐) ๐(๐)(๐ โ ๐) = โ ๐(1 โ ๐) โ Recall exercise 1.17: ๐, ๐ be different distributions on โค, then there exists a distribution ๐ on โค2 such that if (๐) (๐) (๐) (๐, ๐) = ๐, then ๐ = ๐, ๐ = ๐ and ๐[๐ โ ๐] = ๐ ๐๐ (๐, ๐) = 1 โ โ๐ฅโโค min{๐๐ฅ , ๐๐ฅ } Exercise 2.7: Let (๐๐ )๐๐=1 be iid ๐ฅ๐ ~๐ต๐(๐), i.e. ๐[๐๐ = 1] = ๐, ๐[๐๐ = 0] = 1 โ ๐, ๐ โ [0,1], then show that โ๐๐=1 ๐ฅ๐ ~๐ต๐๐(๐, ๐) Exercise 2.8: Let (๐๐ )๐๐=1 be independent, ๐ฅ๐ ~๐๐๐ (๐๐ ), ๐๐ > 0 Show that โ๐๐=1 ๐๐ ~๐๐๐(โ๐๐=1 ๐๐ ) Theorem 2.9: (coupling of Binomial and Poisson random variables) For ๐ > 0, there exists a โค2 values random variable (๐, ๐) such that: ๐ - ๐~๐ต๐๐ (๐, ๐) - ๐~๐๐๐(๐) - ๐[๐ โ ๐] โค ๐2 ๐ Proof: By Ex.1.17, there are iid random variables {(๐ผ๐ , ๐ฝ๐ )}๐๐=1 such that: ๐ - ๐ผ๐ ~๐ต๐ (๐) - ๐ฝ๐ ~๐๐๐ (๐) - ๐[๐ผ๐ = ๐ฝ๐ ] = โ๐ฅโโคโฅ0 min{๐[๐ผ1 = ๐ฅ], ๐[๐ฝ1 = ๐ฅ]} = ๐ ๐ ๐ ๐ ๐ 1โ๐ฅโค๐ โ๐ฅ ๐ ๐ ๐ min {1 โ , ๐ โ๐ } + min { , ๐ โ๐ } = 1 โ + ๐ โ๐ ๐ ๐ ๐ ๐ ๐ Define: ๐ ๐ (๐, ๐) = โ(๐ผ๐ , ๐ฝ๐ ) ๐=1 ๐ By exercise 2.7, ๐~๐ต๐๐ (๐, ๐) By exercise 2.8: ๐~๐๐๐(๐) Finally: ๐ ๐ ๐ ๐ ๐ ๐[๐ โ ๐] โค โ ๐[๐ผ๐ โ ๐ฝ๐ ] = ๐ โ ( โ ๐ โ๐ ) โค ๐ โ ๐ ๐ ๐ ๐=1 ๐ ๐ Exercise 2.10: Let (๐๐ )๐โฅ1 , ๐๐ ~๐ต๐๐ (๐, ) , ๐~๐๐๐(๐), ๐ โฅ 0. (๐) ๐โโ Prove that ๐๐ โ ๐, meaning: For any ๐ด โ โค, ๐[๐๐ โ ๐ด] โ ๐[๐ โ ๐ด] Theorem 2.11 (Poisson and Binomial branching processes): ๐ ๐ Let ๐ and ๐ โ be the total progenies of a ๐ต๐๐ (๐, ) - branching process ๐ and of a ๐๐๐(๐) branching process ๐ โ, ๐ > 0. Then for ๐ โฅ 1 ๐[๐ โฅ ๐] = ๐[๐ โ โฅ ๐] + ๐(๐, ๐) Where |๐(๐, ๐)| โค 2๐2 ๐โ1 โ๐ =1 ๐[๐ โ ๐ > ๐ ] โค 2๐2 ๐ ๐ Proof: By theorem 2.9, we can define the branching processes: (๐1 , ๐2 , โฆ ) (Bin) (๐1โ , ๐2โ , โฆ ) (Poi) (random walk perspective) Such that ๐[๐๐ โ ๐๐โ ] โค ๐2 , ๐ ๐โฅ1 ๐ โฅ 1: {๐ โฅ ๐พ} depends only on {๐1 , โฆ , ๐๐โ1 } - A key observation! Now: ๐โ1 ๐[๐ โฅ ๐, ๐ < ๐] โค โ ๐[๐๐ = ๐๐โ ๐๐๐ 1 โค ๐ โค ๐ โ 1, ๐๐ โ ๐๐ โ ๐ โฅ ๐ โฅ ๐ ] โ ๐ =1 We can deduce from ๐ โฅ ๐ โฅ ๐ that ๐ โ โฅ ๐ ! ๐โ1 โค โ ๐[๐๐ โ ๐๐ โ , ๐ โ โฅ ๐ ] ๐ =1 But these two events are independent ๐โ1 = โ ๐[๐๐ โ ๐ =1 Similarly: ๐โ1 ๐๐ โ ]๐[๐ โ ๐2 โฅ ๐ ] โค โ ๐[๐ โ โฅ ๐ ] ๐ ๐ =1 ๐โ1 ๐[๐ โฅ ๐, ๐ < ๐] โค โ ๐[๐๐ = ๐๐โ , ๐๐๐ ๐ โค ๐ โ 1, ๐๐ โ ๐๐โ , ๐ โ โฅ ๐ โฅ ๐ ] โค โ ๐ =1 ๐โ1 โโ ๐[๐๐ โ ๐๐ โ ] ๐[๐ โ โฅ ๐ ] ๐ =1 โค ๐2 ๐ We are interested in: |๐[๐ โฅ ๐] โ ๐[๐ โ โฅ ๐]| โค |๐[๐ โฅ ๐, ๐ โ โฅ ๐] + ๐[๐ โฅ ๐, ๐ โ < ๐] โ ๐[๐ โ โฅ ๐, ๐ โฅ ๐] โ ๐[๐ โ โฅ ๐, ๐ < ๐]| โค ๐โ1 2๐2 โ ๐[๐ โ โฅ ๐ ] โ ๐ ๐ =1 --- end of lesson 6 Exercise 1.17 ๐, ๐ different distributions on โค (๐๐ฅ โ min{๐๐ฅ , ๐๐ฅ })(๐๐ฆ โ min{๐๐ฆ , ๐๐ฆ }) ๐๐ ๐ฅ = ๐ฆ ๐๐ฅ,๐ฆ = { ๐ ๐๐ (๐, ๐) min{๐๐ฅ , ๐๐ฅ } ๐๐ ๐ฅ = ๐ฆ (๐ฅ, ๐ฆ) โ โค2 1 1 ๐๐๐ (๐, ๐) = โ|๐๐ฅ โ ๐๐ฅ | = ( โ (๐๐ฅ โ ๐๐ฅ ) + โ (๐๐ฅ โ ๐๐ฅ )) = 2 2 ๐ฅโโค ๐ฅ:๐๐ฅ โฅ๐๐ฅ ๐ฅ:๐๐ฅ <๐๐ฅ 1 (โ ๐๐ฅ โ โ ๐๐ฅ โ โ ๐๐ฅ + โ ๐๐ฅ โ โ ๐๐ฅ ) = 2 ๐ฅโโค ๐ฅ:๐๐ฅ <๐๐ฅ ๐ฅ:๐๐ฅ โฅ๐๐ฅ ๐ฅ:๐๐ฅ <๐๐ฅ ๐ฅ:๐๐ฅ <๐๐ฅ 1 (1 โ โ ๐๐ฅ โ โ ๐๐ฅ + โ ๐๐ฅ โ โ ๐๐ฅ โ โ ๐๐ฅ ) = 2 ๐ฅ:๐๐ฅ <๐๐ฅ ๐ฅ:๐๐ฅ โฅ๐๐ฅ ๐ฅโโค ๐ฅ:๐๐ฅ โค๐๐ฅ ๐ฅ:๐๐ฅ <๐๐ฅ 1 (2 โ 2 โ min{๐๐ฅ โ ๐๐ฅ }) = 1 โ โ min{๐๐ฅ , ๐๐ฅ } 2 ๐ฅโโค ๐ฅโโค ๐ is a distribution on โค2 : โ โ ๐๐ฅ,๐ฆ = โ ๐๐ง,๐ง + โ โ ๐๐ฅ,๐ฆ = โ min{๐๐ง , ๐๐ง } + โ(๐๐ฅ โ min{๐๐ฅ , ๐๐ฅ }) = 1 ๐ฅ ๐ฆ ๐งโโค ๐[๐ = ๐ฅ] = โ ๐๐ฅ,๐ฆ ๐ฆโโค ๐ฅโโค ๐ฆโ ๐ฅ ๐งโโค ๐ฅโโค (๐, ๐)~๐ (๐๐ฅ โ min{๐๐ฅ , ๐๐ฅ })๐๐๐ (๐, ๐) = min{๐๐ฅ , ๐๐ฅ } + = ๐๐ฅ ๐๐๐ (๐, ๐) Similarly, ๐[๐ = ๐ฅ] = ๐๐ฅ ๐[๐ โ ๐] = 1 โ ๐[๐ = ๐] = 1 โ โ ๐๐ฅ,๐ฅ = 1 โ โ min{๐๐ฅ , ๐๐ฅ } = ๐๐๐ (๐, ๐) ๐ฅโโค ๐ฅโโค Cannot do better! Assume (๐ โฒ , ๐ โฒ ): ๐ โฒ ~๐, ๐ โฒ ~๐ ๐[๐ โฒ = ๐ โฒ ] = โ ๐[๐ โฒ = ๐ โฒ = ๐ฅ] ๐ฅโโค But ๐[๐ โฒ = ๐ โฒ = ๐ฅ] โค ๐[๐ โฒ = ๐ฅ] = ๐๐ฅ And also ๐[๐ โฒ = ๐ โฒ = ๐ฅ] โค ๐[๐ โฒ = ๐ฅ] = ๐๐ฅ And both are smaller or equal to min{๐๐ฅ , ๐๐ฅ } โ ๐[๐ โฒ โ ๐ โฒ ] = 1 โ ๐[๐ โฒ = ๐ โฒ ] โฅ 1 โ โ min{๐๐ฅ , ๐๐ฅ } = ๐๐๐ (๐, ๐) ๐ฅโโค 3. The Eedos-Renyi random graph Definition 3.1: E.R. random graphs, ๐บ(๐, ๐) for ๐ โฅ 1, ๐ โ [0,1]. Let (๐ผ{๐,๐} )1โค๐<๐โค๐ be iid random variables with distribution ๐ต๐(๐). Then ๐บ(๐, ๐) = ([๐], ๐ธ) Where [๐] = {1, โฆ , ๐}, ๐ธ = {{๐, ๐} โค [๐]: ๐ โ ๐, ๐ผ{๐,๐} = 1} Exercise 3.2: Let ๐ โ number of triangles in ๐บ(๐, ๐) ๐ ๐ธ[๐] = ( ) ๐3 3 1 Let ๐ = ๐๐ผ , ๐ผ > 0 Show that: - ๐โโ For ๐ผ > 1, ๐[๐ > 0] โ ๐โโ For ๐ผ < 1, ๐[๐ > 0] โ ๐ผ = 1? What about ๐พ๐ , ๐ โฅ 4? (hint, prove that ๐ฃ๐๐(๐) ๐โโ โ 0 ๐ธ[๐]2 0 (can be proven using a chebishev inequality) 1 (should use a second moment method! โ the variant of ๐) and use chebishev). Notation: - For ๐ข, ๐ฃ โ [๐], "๐ข โ ๐ฃโ means โThere is a nearest-neighbor path from ๐ข to ๐ฃ in ๐บ(๐, ๐)โ - For ๐ฃ โ [๐], ๐ถ(๐ฃ) = {๐ค โ [๐]: ๐ฃ โ ๐ค} Consider the following exploration algorithm for ๐ถ(1) set ๐ด0 = {1}, ๐1 = [๐]\{1} - For ๐ก > 0, if ๐ด๐กโ1 = โ , then ๐ด๐ก = โ , ๐๐ก = โ (terminate) If ๐ด๐กโ1 โ โ , let ๐ค๐กโ1 be the smallest vertex in ๐ด๐กโ1 and define: ๐ฃ๐ก = {๐ค โ ๐๐กโ1 |{๐ค๐กโ1 , ๐ค} โ ๐ธ} ๐ด๐ก โ (๐ด๐กโ1 \{๐ค๐กโ1 }) โช ๐๐ก ๐๐ก โ ๐๐กโ1 \๐๐ก Example: Draw graphical example. Note that: - |๐ด0 | = 1, |๐ด๐ก | = |๐ด๐กโ1 | + |๐๐ก | โ 1 for ๐ก โฅ 1 - |๐๐ก | = ๐ โ |๐ด๐ก | โ ๐ก, ๐ก โฅ 0 - |๐ถ(1)| = inf{๐ก โฅ 1 | |๐ด๐ก | = 0} Lemma 3.3: For ๐ฅ1 , โฆ , ๐ฅ๐ โ โคโฅ0 , let: - ๐ 0 = 1, ๐ ๐ก = ๐ ๐กโ1 + ๐ฅ๐ก โ 1 for ๐ก โฅ 1 - ๐๐ก = ๐ โ ๐ ๐ก โ ๐ก If ๐ 1 , ๐ 2 , โฆ , ๐ ๐ โฅ 1, ๐0 , ๐1 , โฆ , ๐๐โ1 โฅ 0 Then ๐ ๐[|๐1 | = ๐ฅ1 , โฆ , |๐๐ | = ๐ฅ๐ ] = โ(๐(๐๐กโ1 , ๐, ๐ฅ๐ก )) ๐ Where ๐(๐, ๐, ๐) = ( ) ๐๐ (1 โ ๐)๐โ๐ ๐ ๐ก=1 Proof: By picture. TODO: Draw picture proof. ๐[|๐1 | = ๐ฅ1 , โฆ , |๐๐ | = ๐ฅ๐ ] = โ โ ๐ [|๐1 | = ๐ฅ1 , โฆ , |๐๐โ1 | = ๐ฅ๐โ1 , ๐ด๐โ1 = ๐ด, ๐๐โ1 = ๐, โ ๐ผ{๐,๐ค} = ๐ฅ๐ ] ๐ดโ[๐] ๐โ[๐] ๐ดโ โ ๐คโ๐ ๐ is the smallest vertex in ๐ด. {|๐1 } = ๐ฅ1 , โฆ , ๐๐กโ1 = ๐} โ ๐(๐ผ๐ : ๐ ๐๐๐ก ๐๐ ๐๐๐๐ ๐๐๐๐ ๐ ๐ก๐ ๐) { โ ๐ผ{๐,๐ค} = ๐ฅ๐ } โ ๐(๐ผ๐ : ๐ ๐๐๐๐ ๐๐๐๐ {๐} ๐ก๐ ๐) ๐คโ๐ So these two events must be independent! So: ๐[|๐1 | = ๐ฅ1 , โฆ , |๐๐ | = ๐ฅ๐ ] = โ โ ๐[|๐1 | = ๐ฅ1 , โฆ , ๐๐โ1 = ๐] โ ๐(๐๐โ1 , ๐, ๐ฅ๐ ) If the probability of ๐[|๐1 | = ๐ฅ1 , โฆ , ๐๐โ1 = ๐] is positive then necessarily |๐| = |๐๐โ1 | = ๐๐โ1 โ ๐[|๐1 | = ๐ฅ1 , โฆ , |๐๐ | = ๐ฅ๐ ] = ๐[๐1 = ๐ฅ! , โฆ , |๐๐โ1 | = ๐ฅ๐โ1 ] × ๐(๐๐โ1 , ๐, ๐ฅ๐ ) So now we can use induction to complete the proof. --- end of lesson Definition: 3.4: ๐ ๐กโ1 Let (๐ผ๐ก,๐ )๐ก,๐โฅ1 be iid ๐ต๐(๐). Define ๐0 = ๐ โ 1, for ๐ก โฅ 1 : โ๐=1 ๐ผ๐ก,๐ , ๐๐ก = ๐๐ก โ ๐๐ก , ๐๐ก = (๐1 + โฏ + ๐๐ก ) โ (๐ก โ 1) ๐๐ก ~|๐๐ก |, ๐๐ก ~|๐๐ก |, ๐๐ก ~|๐ด๐ก | Lemma 3.5 For ๐ฅ๐ , ๐ ๐ , ๐๐ as in lemma 3.3, ๐ โ ๐(๐๐กโ1 , ๐, ๐ฅ๐ก ) , ๐[๐1 = ๐ฅ1 , โฆ , ๐๐ = ๐ฅ๐ ] = { ๐๐ ๐1 , โฆ , ๐๐กโ1 โฅ 0 ๐ก=0 0, ๐๐กโ๐๐๐ค๐๐ ๐ Proof: {๐1 = ๐ฅ1 , โฆ , ๐๐ = ๐ฅ๐ } โ {๐1 = ๐1 , โฆ , ๐๐โ1 = ๐๐โ1 } So if ๐๐ < 0, then ๐[โฆ ] = 0. Otherwise: ๐[๐1 = ๐ฅ1 , โฆ , ๐๐โ1 = ๐ฅ๐โ1 , ๐๐ = ๐ฅ๐ ] = ๐[๐1 = ๐ฅ1 , โฆ , ๐๐โ1 = ๐ฅ๐โ1 ] โ ๐(๐๐โ1 , ๐, ๐ฅ๐ ) ๐๐ฆ ๐๐๐ ๐๐โ1 Because ๐๐ = โ๐=1 ๐ผ๐,๐ We can continue by induction. And that finishes the proof. โ Corollary 3.6: For ๐ โฅ 1, and any ๐: โค๐โฅ0 โ โ ๐ธ[๐(|๐1 |, โฆ , |๐๐ |)1{|๐ด1 |>0,โฆ,|๐ด๐โ1 |>0} ] = ๐ธ[๐(๐1 , โฆ , ๐๐ )1{๐1 >0,โฆ,๐๐โ1 >0} ] Proof: Combine Lemma 3.3 and Lemma 3.5. โ Theorem 3.7: Let ๐ โ and ๐ + be the total progenies of a ๐ต๐๐(๐ โ ๐, ๐) and ๐ต๐๐(๐, ๐) branching processes. Where 1 โค ๐ โค ๐ and ๐ โ [0,1] then ๐[๐ โ โฅ ๐] โค ๐[|๐ถ(1)| โฅ ๐] โค ๐[๐ + โฅ ๐]. +/โ Proof: For (๐ผ๐ก,๐ ) as in Definition 3.4, define ๐๐ก +/โ Then (๐๐ก ) are iid~๐ต๐๐(๐/๐ โ ๐, ๐). ๐กโฅ1 +/โ +/โ +/โ ๐๐ก : = (๐1 + โฏ + ๐๐ก ) โ (๐ก โ 1), (๐) +/โ Then ๐ +/โ = inf {๐ก โฅ 1|๐๐ก = 0} ๐/๐โ๐ = โ๐=1 ๐ก โฅ 1, ๐ผ๐ก,๐ , ๐ก โฅ 1. +/โ ๐0 =1 ๐[|๐ถ(1)| โฅ ๐] = ๐[|๐ด1 | > 0, โฆ , |๐ด๐โ1 | > 0] = ๐[๐1 > 0, โฆ , ๐๐โ1 > 0] โค + ๐[๐1+ > 0, โฆ , ๐๐โ1 > 0] = ๐[๐ + โฅ ๐] (by cor 3.6 if ๐ โก 1) ๐๐ก โค ๐๐ก+ , ๐ก โค ๐ โ 1 On the other hand, ๐[|๐ถ(1)| โค ๐] = ๐[โ๐๐ก=1{|๐๐ก | = 0, |๐ด๐กโ1 | = 1, |๐ด๐กโ2 | > 0, โฆ , |๐ด1 | > 0}] (exploration stops at ๐ก) ๐ = โ ๐[๐๐ก = 0, ๐๐กโ1 = 1, ๐๐กโ2 > 0, โฆ , ๐1 > 0] ๐ก=1 (by Cor 3.6) = ๐[๐ โค ๐] where ๐ = inf{๐ก โฅ 1|๐๐ก = 0} Claim: If {๐ โค ๐} โ {๐ โ โค ๐} Indeed, if ๐ โค ๐ ๐๐ = ๐ โ 1 โ ๐1 โ โฏ โ ๐๐ = ๐ โ ๐ โ ๐๐ โฅ ๐ โ ๐ So ๐1 โฅ ๐2 โฅ โฏ โฅ ๐๐ โฅ ๐ โ ๐ Hence, ๐๐กโ โค ๐๐ก , ๐ก โค ๐ ๐๐กโ โค ๐๐ก , ๐ก โค ๐ โ ๐โ โค ๐ Which proves this claim. โ Exercise 3.8: Consider random variables ๐, ๐ such that ๐~๐ต๐๐(๐, ๐) and conditionally on ๐, ๐~๐ต๐๐(๐, ๐), ๐ โฅ 1, ๐, ๐ โ [0,1] ๐ I.e. ๐[๐ = ๐|๐ = ๐] = ( ) ๐ ๐ (1 โ ๐)๐โ๐ ๐ (0 โค ๐ โค ๐ โค ๐) Prove that ๐~๐ต๐๐(๐, ๐๐) It would be probable to assume: |๐๐ก |~๐ต๐๐(๐ โ 1, (1 โ ๐)๐ก ) But this is not correctโฆ Proposition 3.9: For 0 โค ๐ก โค ๐, ๐๐ก ~๐ต๐๐(๐ โ 1, (1 โ ๐)๐ก ) Proof: By induction. ๐0 = ๐ โ 1. This is true. Assume that ๐๐ก ~๐ต๐๐(๐ โ 1, (1 โ ๐)๐ก ) Need to check that conditionally on ๐๐ก , ๐๐ก+1 ~๐ต๐๐(๐๐ก , 1 โ ๐) (By Ex 3.6) 0 โค ๐ โค ๐ โค ๐ โ 1 ๐[๐๐ก+1 = ๐. ๐๐ก = ๐], ๐๐ก+1 = ๐๐ก โ ๐๐ก+1 = ๐[ ๐ โ ๐ก+1 = ๐ โ ๐, ๐๐ก = ๐] ๐๐๐๐๐๐๐๐๐๐๐๐ = ๐(๐, ๐, ๐ โ ๐)๐[๐๐ก = ๐] = =โ๐ ๐=1 ๐ผ๐ก+1,๐ ๐(๐, (1 โ ๐), ๐)๐[๐๐ก = ๐] โ ๐ ( ) ๐๐โ๐ (1 โ ๐)๐ ๐โ๐ ๐ผ๐ = ๐ โ 1 โ log ๐ , (see Ex. 1.16, ๐ผ(๐)) ๐>0 ๐ Lemma 3.8: For ๐ = ๐ , ๐ โ (0,1) ๐ โฅ 1, ๐[|๐ถ(1)| > ๐] โค ๐ โ๐ผ๐ ๐ Proof: Let ๐ > 0, ๐ = โ log ๐ > 0 ๐[|๐ถ(1)| > ๐] = ๐[|๐ด1 | > 0, โฆ , |๐ด๐ | > 0] ๐ถ๐๐ 3.6 = โค ๐[๐1 > 0, โฆ , ๐๐ > 0] โค ๐[๐๐ > 0] = ๐ ๐ โ๐ + ๐ โ 1 โฅ ๐ ๐๐๐ข๐๐ฃ๐๐๐๐๐ก ๐ก๐: [ ] ๐๐ + ๐ โ 1 = ๐1 + โฏ + ๐๐ = ๐ โ 1 โ ๐๐ ~๐ต๐๐(๐ โ 1,1 โ (1 โ ๐)๐ ) ๐ ๐(๐๐ +๐โ1) โฅ๐ ๐๐ So ๐[|๐ถ(1)| > ๐] โค ๐[๐ ๐(๐๐ +๐โ1) โฅ ๐ ๐๐ ] ๐๐ฆ ๐๐๐๐๐๐ฃ โค ๐ โ๐๐ ๐ธ[๐ ๐(๐๐ +๐โ1) ] ๐โ1 ๐ธ[๐ ๐(๐๐ +๐โ1) ] = ((1 โ ๐) + ๐๐ ๐ ) (๐) ๐๐ + ๐ โ 1 = ๐1 + โฏ + ๐๐โ1 , ๐๐ iid ~๐ต๐(๐) ๐โ1 โ ๐ธ[๐ ๐(๐๐ +๐โ1) ] = ๐ธ [๐ ๐(โ๐โ1 ๐=1 ๐๐ ) ๐โ1 ] = โ(๐ธ[๐ ๐๐๐ ]) = (๐ธ[๐ ๐๐1 ]) ๐โ1 = ((1 โ ๐) + ๐๐ ๐ ) ๐=1 So ๐[|๐ถ(1)| > ๐] โค ๐ โ๐๐ ๐ ๐โ1 ((1 โ ๐) + ๐๐ ) (1โ๐ฅ)โค๐ โ๐ฅ โค ๐ exp(โ๐๐ + ๐(๐ โ 1)(๐ โ 1)) โค exp(โ๐๐ + ๐(๐ ๐ โ 1)๐) ๐ = 1 โ (1 โ ๐)๐ โค ๐๐ (โ) (comment: Seems like ๐ก turned into ๐) โ ๐[|๐ถ(1)| > ๐] โค exp(โ๐๐ + ๐(๐ ๐ โ 1)๐) โค exp(โ๐๐ + ๐๐(๐ ๐ โ 1)๐) = ๐ ๐= ๐ ๐ exp (โ๐(๐ โ ๐๐(๐ โ 1))) = exp (โ๐(๐ โ ๐(๐ ๐ โ 1))) If we try to minimize the expression, we get that ๐ = โ log ๐ So letโs just set it in: ๐ โ๐๐ผ๐ Why is (*) true? Show that at zero they are equal, and the derivative of the right hand side is always larger then the derivative of the left hand side. --- end of lesson Theorem 3.9 (Upper bound on the largest sub-critical component): ๐ 1 Fix ๐ โ (0,1), ๐ = ๐ , ๐ > ๐ผ ๐ Then there is a ๐ฟ = ๐ฟ(๐, ๐) such that ๐[|๐ถ๐๐๐ฅ | > ๐ log ๐] < ๐โ๐ฟ Proof: ๐[|๐ถ๐๐๐ฅ | > ๐ log ๐] = ๐ [ โ {|๐ถ(๐ฃ)| > ๐ log ๐}] โค โ ๐[|๐ถ(๐ฃ)| > ๐ log ๐] = ๐ฃโ[๐] ๐๐ฆ ๐ฟ๐๐๐๐ 3.8 ๐๐[|๐ถ(1)| > ๐ log ๐] โค ๐ฃโ[๐] ๐ โ ๐โ๐ผ๐๐ , ๐ผ๐ ๐ > 1 = โ Next, lower bound: Lemma 3.10: Let ๐โฅ๐ = โ๐๐ฃ=1 1{|๐ถ(๐ฃ)|โฅ๐} ๐โฅ๐ = ๐ธ[|๐ถ(1)|1{|๐ถ(1)|โฅ๐} ] Then ๐ฃ๐๐(๐โฅ๐ ) โค ๐๐โฅ๐ 2 2 Proof: ๐ฃ๐๐(๐โฅ๐ ) = ๐ธ [(โ๐๐ฃ=1 1{|๐ถ(๐ฃ)|โฅ๐} ) ] โ ๐ธ[โ๐๐ฃ=1 1{|๐ถ(๐ฃ)|โฅ๐} ] = ๐ ๐ โ โ ๐[|๐ถ(๐)| โฅ ๐, |๐ถ(๐)| โฅ ๐] โ ๐[|๐ถ(๐)| โฅ ๐]๐[|๐ถ(๐)| โฅ ๐] ๐=1 ๐=1 Split according to ๐ โ ๐ and ๐ โ ๐ ๐ connected to ๐ and ๐ not connected to ๐ ๐ ๐ ๐[|๐ถ(๐)| โฅ ๐, |๐ถ(๐)| โฅ ๐, ๐ โ ๐] = โ โ ๐[|๐ถ(๐)| = ๐, ๐ โ ๐, |๐ถ(๐)| = ๐] = ๐=๐ ๐=๐ ๐ โ ๐[|๐ถ(๐) = ๐|, ๐ โ ๐] โ ๐ [|๐ถ(๐)| = ๐] = โ ๐[|๐ถ(๐)| = ๐, ๐ โ ๐] โ ๐ [|๐ถ(๐)| โฅ ๐] โค (๐โ๐) ๐,๐ ๐ (๐โ๐) ๐=๐ ๐ โ ๐[|๐ถ(๐)| = ๐, ๐ โ ๐] โ ๐ [|๐ถ(๐)| โฅ ๐] = โ ๐[|๐ถ(๐)| = ๐] โ ๐[|๐ถ(๐)| โฅ ๐] (๐โ๐) ๐=๐ ๐=๐ (we can strike these out since it only increases the probability of the event) So letโs go back to the entire variance: The term we just calculated cancels the second term! So we are only left with the case where they are connected: ๐ ๐ ๐ฃ๐๐(๐โฅ๐ ) โค โ โ ๐[|๐ถ(๐)| โฅ ๐, ๐ถ(๐) โฅ ๐, ๐ โ ๐] ๐=1 ๐=1 ๐โ๐โ ๐โ๐๐ ๐ ๐ ๐๐๐๐๐๐ฆ ๐ ๐กโ๐ ๐ ๐๐๐ ๐๐ฃ๐๐๐ก = ๐ โ โ ๐[|๐ถ(๐)| โฅ ๐] = ๐=1 ๐=1 ๐ ๐ โ ๐ธ 1{|๐ถ(๐)|โฅ๐} โ โ 1{๐โ๐ถ(๐)} = ๐๐โฅ๐ ๐=1 ๐=1 โ [ ] =|๐ถ(๐)| Theorem 3.11 (lower bound on largest subcritical component): ๐ Fix ๐ โ (0,1), ๐ = ๐ 1 Then for every ๐ < ๐ผ there exists ๐ฟ = ๐ฟ(๐, ๐), and ๐ = ๐(๐, ๐) such that ๐ ๐[|๐ถ๐๐๐ฅ | โค ๐ log ๐] โค ๐๐โ๐ฟ Proof: Let ๐ = โ๐ log ๐โ, ๐ denotes a constant depending only on ๐, ๐ changing from place to place. Claim 1: For any 0 < ๐ผ < 1 โ ๐ผ๐ ๐ ๐ธ[๐โฅ๐ ] โฅ ๐๐ผ , for ๐ โฅ ๐ Claim 2: ๐ฃ๐๐(๐โฅ๐ ) โค ๐๐๐1โ๐๐ผ๐ From these claims, it follows that we can prove this theorem. Why? ๐[|๐ถ๐๐๐ฅ | โค ๐ log ๐] = ๐[๐โฅ๐ = 0] โค ๐[|๐๐โฅ0 โ ๐ธ[๐โฅ๐ ]| โฅ ๐ธ[๐โฅ๐ ]] = ๐๐ฆ ๐กโ๐ ๐[|๐๐โฅ0 โ ๐ธ[๐โฅ๐ ]|2 โฅ ๐ธ[๐โฅ๐ ]2 ] ๐ฃ๐๐(๐โฅ๐ ) ๐๐๐๐๐๐ ๐๐๐1โ๐๐ผ๐ โค โค ๐ธ[๐โฅ๐ ]2 ๐2๐ผ 2 (For instance, if ๐ผ = 3 (1 โ ๐๐ผ๐ ) then itโs โค ๐ log ๐ ๐ (1โ๐๐ผ๐ ) 3 ) Letโs prove claim 1: ๐ Let ๐, ๐ โ be the total progenies of ๐ต๐๐ (๐ โ ๐, ๐) branching process, and ๐๐๐(๐๐ ) branching process wher ๐๐ = ๐(๐โ๐) ๐ By theorem 3.7, ๐[|๐ถ(1)| โฅ ๐] โฅ ๐[๐ โฅ ๐] ๐๐ฆ ๐กโ๐๐๐๐๐ 2.11 โฅ ๐ 1 ๐(๐ โ ๐) = ( ) ๐ ๐โ๐โ ๐ ๐๐ โ โ ๐[๐ โฅ ๐] โฅ ๐[๐ = ๐] ๐๐ฆ ๐กโ๐๐๐๐๐ 2.3 = We can use sterlingโs formula: (๐๐ ๐)๐โ1 โ๐ ๐ ๐ ๐ ๐! ๐[๐ โ โฅ ๐] โ 2๐2๐ ๐ ๐ ๐ ๐ ๐! = ( ) โ โ2๐๐(1 + ๐(1)) ๐ And then: โฅ ๐๐๐ ๐ โ๐๐ ๐ ๐ ๐ (1 + ๐(1)) โฅ ๐ โ๐ผ๐๐ ๐(1+๐) ๐๐๐ ๐ ๐ข๐๐๐๐๐๐๐๐ก๐๐ฆ ๐๐๐๐๐ ๐ ๐โ2๐๐๐๐ (๐ผ๐ = ๐ โ 1 โ log ๐) = ๐ โ๐ผ๐ ๐(1+๐) For ๐ โฅ 0: 1 โ ๐๐ผ๐ (1 + ๐) > ๐ผ for ๐ โฅ ๐(๐, ๐, ๐) >๐ผ โ ๐ธ[๐โฅ๐ ] = ๐๐[|๐ถ(1)| โฅ ๐] โฅ ๐๐[๐ โฅ ๐] โ ๐๐ ๐ โฅ ๐ (๐ โ๐ผ๐ ๐(1+๐) โ ๐๐ ) ๐ =๐ โ 1โ๐ผ๐ ๐(1+๐) โ ๐ log ๐ โฅ ๐๐ผ for ๐ โฅ ๐. (For ๐ chosen sufficiently small) Now letโs prove claim 2: Proof of claim 2: |๐ถ(1)| We can write |๐ถ(1)| = โ๐=1 1{|1|>๐} = โ๐๐=1 1{|1|>๐} So: ๐ ๐ ๐โฅ๐ = โ ๐[|๐ถ(1)| โฅ 1, |๐ถ(1)| โฅ ๐] = ๐๐[|๐ถ(1)| โฅ ๐] + โ ๐[|๐ถ(1)| โฅ ๐] ๐=1 ๐=๐+1 โ ๐๐ โ๐ผ๐ (๐โ1) + โ ๐ โ๐ผ๐ (๐โ1) โค ๐๐๐โ๐๐ผ๐ + ๐๐ โ โ๐ผ๐ ๐ โค ๐๐๐โ๐๐ผ๐ โค๐โ๐๐ผ๐ ๐=๐+1 So by lemma 3.10 get claim 2. โ Exercise 3.12 (second moment method): Let ๐ be a random variable such that 0 โค ๐ธ[๐] < โ and 0 < ๐ธ[๐ 2 ] < โ Let 0 โค ๐ < 1. ๐ธ[๐ฅ]2 Prove that: ๐[๐ > ๐๐ธ[๐]] โฅ (1 โ ๐)2 ๐ธ[๐ฅ 2 ] In our proof, we used it with ๐ = 0. Hint: Prove first that (1 โ ๐)๐ธ[๐] โค ๐ธ[๐ โ 1{๐>๐๐ธ[๐]} ] ๐๐ฆ ๐๐๐๐๐ 3.8 โค
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