Method of cumulants and mod-Gaussian convergence of the graphon - PowerPoint PPT Presentation

Method of cumulants and mod-Gaussian convergence of the graphon models Pierre-Loïc Méliot (Joint work with Valentin Féray and Ashkan Nikeghbali) 2017, May 11th University Paris-Sud

1 This is not the whole story: 2 S n 1 S n When looking at a sum S n = ∑ n i = 1 A i of centered i.i.d. random variables, the fluctuations are universally predicted by the central limit theorem √ ⇀ N ( 0 , 1 ) . n Var ( A 1 ) ▶ large deviations (Cramér, 1938): log ( P [ S n ≥ nx ]) ≃ − n I ( x ) . ▶ speed of convergence (Berry, 1941; Esseen, 1945): � � [ ] ∫ s � � 3 E [ | A 1 | 3 ] � � e − t 2 √ ≤ s − √ � ≤ ( Var ( A 1 )) 3 / 2 √ n . sup � P � � 2 dt n Var ( A 1 ) 2 π s ∈ R −∞ ▶ local limit theorem (Gnedenko, 1948; Stone, 1965): if A 1 is non- lattice distributed and Var ( A 1 ) = 1, then √ √ √ [ ] ≃ e − x 2 √ n P S n ∈ ( nx , nx + h ) h . 2 π

Many other sequences of random variables are asymptotically normal: functionals of Markov chains, martingales, etc. 6 S n 2 Definition (Mod-Gaussian convergence) time the CLT and the other limiting results. lows one to go beyond the central limit theorem, and to prove in one Idea: there is a renormalisation theory of random variables that al- 2 A sequence of real random variables ( X n ) n ∈ N is mod-Gaussian with parameters t n → + ∞ and limit ψ ( z ) if, locally uniformly on a domain D ⊂ C , E [ e zX n ] e − tnz 2 = ψ n ( z ) → ψ ( z ) with ψ continuous on D and ψ ( 0 ) = 1. n 1 / 3 ; t n = n 1 / 3 Var ( A 1 ) and For a sum of i.i.d. S n , one looks at X n = ψ ( z ) = exp( E [( A 1 ) 3 ] z 3 ) .

3 Objectives: of random graphs. 4 3. Prove the mod-Gaussian convergence of a large class of models vergence. 2. Describe general conditions which ensure the mod-Gaussian con- 1. Explain the consequences of mod-Gaussian convergence. Later: Markov chains, random graphs, random permutations, etc. has the mod-Gaussian convergence Example: let X n = Re (log det( I n − M n )) , with M n ∼ Haar ( U ( n )) . One → G ( 1 + z 2 ) 2 E [ e zX n ] e − (log n ) z 2 G ( 1 + z ) , G = Barnes’ function . Remark: one can replace the exponent z 2 2 of the Gaussian distribution by the exponent η ( z ) of any infinitely divisible distribution.

Mod-Gaussian convergence and bounds on cumulants

Method of cumulants If X is a random variable with convergent Laplace transform, its cumu- Idea: characterize similarly the mod-Gaussian convergence of a se- z r . The first cumulants are 4 dz r )� lants are: ( � κ ( r ) ( X ) = d r � log E [ e zX ] . � z = 0 So, log E [ e zX ] = ∑ ∞ κ ( r ) ( X ) r = 1 r ! κ ( 2 ) ( X ) = E [ X 2 ] − ( E [ X ]) 2 = Var ( X ) κ ( 1 ) ( X ) = E [ X ] ; ; κ ( 3 ) ( X ) = E [ X 3 ] − 3 E [ X 2 ] E [ X ] + 2 ( E [ X ]) 3 . The Gaussian distribution N ( m , σ 2 ) is characterized by κ ( 1 ) ( X ) = m , κ ( 2 ) ( X ) = σ 2 , κ r ≥ 3 ( X ) = 0. quence ( X n ) n ∈ N .

Definition (Method of cumulants) (MC2) The first cumulants satisfy (MC3) All the cumulants satisfy 5 A sequence of random variables ( S n ) n ∈ N satisfies the hypotheses of the method of cumulants with parameters ( D n , N n , A ) if: (MC1) One has N n → + ∞ and D n N n → 0. κ ( 1 ) ( S n ) = 0 ; κ ( 2 ) ( S n ) = ( σ n ) 2 N n D n ; κ ( 3 ) ( S n ) = L n N n ( D n ) 2 with lim n →∞ ( σ n ) 2 = σ 2 > 0 and lim n →∞ L n = L . | κ ( r ) ( S n ) | ≤ N n ( 2 D n ) r − 1 r r − 2 A r .

Mod-Gaussian convergence and its consequences Lz 3 dx This inequality relies on the general estimate N n D n S n Consequences: . 6 6 S n D n N n If ( S n ) n ∈ N satisfies the hypotheses MC1-MC3, then X n = ( N n ) 1 / 3 ( D n ) 2 / 3 is mod-Gaussian convergent, with t n = ( σ n ) 2 ( ) 1 / 3 ( ) and ψ ( z ) = exp √ Var ( S n ) , then Y n ⇀ N ( 0 , 1 ) . 1. Central limit theorem: if Y n = 2. Speed of convergence: ) 3 √ ( 3 A d Kol ( Y n , N ( 0 , 1 )) ≤ . σ n � � � � ∫ T � � � � � µ ( ξ ) − � ν ( ξ ) d ν ( x ) � � � � d Kol ( µ , ν ) ≤ 1 � d ξ + 24 . � � � π ξ π T − T ∞

7 D n D n ment of change of measure. Ly 3 D n N n 2 N n y 2 , then This estimate relies on the Berry–Esseen inequality and an argu- and D n N n N n D n . , then D n N n N n ( ) 1 / 6 3. Normality zone and moderate deviations: if y ≪ P [ Y n ≥ y ] = P [ N ( 0 , 1 ) ≥ y ] ( 1 + o ( 1 )) . ( ) 1 / 4 If 1 ≪ y ≪ ( ) √ P [ Y n ≥ y ] = e − y 2 √ exp ( 1 + o ( 1 )) . 6 σ 3 2 π 4. Local limit theorem: for any exponent ε ∈ ( 0 , 1 2 ) , ( N n ) ε [ ( D n ) ε ] = e − y 2 Y n − y ∈ √ ( b − a ) . lim P ( a , b ) n →∞ 2 π ( ) − 1 / 2 ( ) 1 / 6 Thus, Y n is normal between the two scales

Joint cumulants and dependency graphs

Dependency graphs 2 7 6 5 4 3 1 Example: 8 Let S = ∑ v ∈ V A v be a sum of random variables, and G = ( V , E ) a de- pendency graph for ( A v ) v ∈ V : if V 1 and V 2 are two disjoint subsets of V without edge e = { v 1 , v 2 } between v 1 ∈ V 1 and v 2 ∈ V 2 , then ( A v ) v ∈ V 1 and ( A v ) v ∈ V 2 are independent. ( A 1 , A 2 , . . . , A 5 ) ⊥ ( A 6 , A 7 ) , but one has also ( A 1 , A 2 , A 3 ) ⊥ A 5 . Parameters of the graph: D = max v ∈ V (deg v + 1 ) , N = card ( V ) , A = max v ∈ V ∥ A v ∥ ∞ .

Theorem (Bound on cumulants; Féray–M.–Nikeghbali, 2013) d r A j If S is a sum of random variables with a dependency graph of param- 9 eters ( D , N , A ) , then for any r ≥ 1, | κ ( r ) ( S ) | ≤ N ( 2 D ) r − 1 r r − 2 A r . Corollary: if S n = ∑ N n i = 1 A i , n with the A i , n ’s bounded by A and a sparse dependency graph of maximal degree D n ≪ N n , then MC3 is satisfied. The proof of the bound relies on the notion of joint cumulant : )� � ( � log E [ e z 1 A 1 + z 2 A 2 + ··· + z r A r ] κ ( A 1 , A 2 , . . . , A r ) = � dz 1 dz 2 · · · dz r z 1 = ··· = z r = 0   ℓ ( π ) ∑ ∏ ∏ ( − 1 ) ℓ ( π ) − 1 ( ℓ ( π ) − 1 )!  . = E π 1 ⊔ π 2 ⊔···⊔ π ℓ ( π ) =[ [ 1 , r ] ] i = 1 j ∈ π i

Properties of joint cumulants 10 2. The joint cumulants are multilinear and invariant by permutation. 1. For any random variable X , κ ( r ) ( X ) = κ ( X , X , . . . , X ) ( r occurrences). 3. If { A 1 , A 2 , . . . , A r } can be split in two independent families, then κ ( { A 1 , . . . , A r } ) = 0. Consider a sum S = ∑ v ∈ V A v with a dependency graph G of parameters ( D , N , A ) . ∑ κ ( r ) ( S ) = κ ( A v 1 , A v 2 , . . . , A v r ) v 1 , v 2 ,..., v r and the sum can be restricted to families { v 1 , v 2 , . . . , v r } such that the induced multigraph H = G [ v 1 , v 2 , . . . , v r ] is connected. Actually, | κ ( A v 1 , A v 2 , . . . , A v r ) | ≤ A r 2 r − 1 ST H , where ST H is the number of spanning trees of H .

Sketch of proof of the bound the bivariate Tutte polynomial 11 1. In the expansion of κ ( A 1 , . . . , A r ) , many set partitions yield the same moment M π = ∏ ℓ ( π ) i = 1 E [ ∏ j ∈ π i A j ] , so ( ∑ ) ∑ κ ( A 1 , . . . , A r ) = µ ( π ) M π ′ π ′ π → H π ′ � � � � | κ ( A 1 , . . . , A r ) | ≤ A r ∑ ∑ � � � µ ( π ) � � . � π ′ π → H π ′ 2. The functional F H /π ′ = ∑ π → H π ′ µ ( π ) depends only on the con- traction H /π ′ of H along π ′ , and one can show that is up to a sign | F H /π ′ | = T H /π ′ ( 1 , 0 ) ≤ T H /π ′ ( 1 , 1 ) = ST H /π ′ .

tree of H , hence The bound on the cumulant of the sum S follows by noticing that: 12 3. A pair ( π ′ , T ∈ ST H /π ′ ) can be associated to a bicolored spanning ∑ ST H /π ′ ≤ 2 r − 1 ST H . π ′ ▶ given a vertex v 1 and a Cayley tree T , the number of lists ( v 2 , . . . , v r ) such that T is contained in H = G [ v 1 , . . . , v r ] is smaller than D r − 1 ; ▶ the number of pairs ( v 1 ∈ V , T Cayley tree ) is N r r − 2 . The proof leads to the notion of weighted dependency graph .

Weighted dependency graphs Definition (Weighted dependency graph; Féray, 2016) The same proof gives: 13 A sum S = ∑ v ∈ V A v admits a weighted dependency graph G = ( V , E ) of parameters ( wt : E → R + , A ) if, for any family { v 1 , v 2 , . . . , v r } ,   ∑ ∏   . | κ ( A v 1 , A v 2 , . . . , A v r ) | ≤ A r wt ( v i , v j ) T ∈ ST G [ v 1 ,..., vr ] ( v i , v j ) edge of T | κ ( S ) | ≤ N ( 2 D ) r − 1 r r − 2 A r 2 ( 1 + max v ∈ W ( ∑ with N = card ( V ) and D = 1 w ∼ v wt ( v , w ))) .

Sums of weakly dependent random variables N n and all its consequences. Moreover, one has the concentration in- N n D n equality: D n and a dependency graph of maximal degree D n . We suppose that 14 Let S n = ∑ N n i = 1 A i , n be a sum of random variables, with | A i , n | ≤ A a.s. κ ( 3 ) ( S n ) Var ( S n ) → σ 2 > 0 → 0 ; ; N n ( D n ) 2 → L . Then, S n − E [ S n ] satisfies the hypotheses of the method of cumulants, ( ) ε 2 P [ | S n − E [ S n ] | ≥ ε ] ≤ 2 exp − 9 ( ∑ N n i = 1 E [ | A i | ]) D n A ( ) ε 2 ≤ 2 exp − . 9 N n D n A 2