Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Extreme Value Theory with Operator Norming Stilian Stoev ( sstoev@umich.edu ) University of Michigan, Ann Arbor Nov 9, 2012 Workshop on Spatial Extreme Value Theory and Properties of Max–Stable Processes In honor of the habilitation of Professeur Cl´ ement Dombry Joint work with Mark M. Meerschaert and Hans-Peter Scheffler.
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Some Limit Theory 1 Representation and Simulation 2 Testing for Hetero-Ouracity 3 Implementation and Applications 4 References 5
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Operator Normalized Exteremes
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Preliminaries A r.vector X in R d has an operator regularly varying law if in R d \ { 0 } v nP { A n X ∈ ·} − → φ ( · ) , (1) for a sequence of d × d matrices A n → 0. To avoid trivialities the limit measure φ is assumed to be full, i.e. it’s support is not concentrated on a sub-space of R d As shown in Ch. 6.1 of Meerschaert & Scheffler (2001), for all B ∈ B ( R d \ { 0 } ) , t > 0 , t φ ( B ) = φ ( t − E B ) , where t − E = e − log( t ) E = � ∞ n =0 ( − log ( t )) n E n / n !.
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Preliminaries A r.vector X in R d has an operator regularly varying law if in R d \ { 0 } v nP { A n X ∈ ·} − → φ ( · ) , (1) for a sequence of d × d matrices A n → 0. To avoid trivialities the limit measure φ is assumed to be full, i.e. it’s support is not concentrated on a sub-space of R d As shown in Ch. 6.1 of Meerschaert & Scheffler (2001), for all B ∈ B ( R d \ { 0 } ) , t > 0 , t φ ( B ) = φ ( t − E B ) , where t − E = e − log( t ) E = � ∞ n =0 ( − log ( t )) n E n / n !. Moreover, the A n ’s can be chosen so that A [ tn ] A − 1 → t − E , n → ∞ . − n
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Examples scalar: E = 1 /α I d = diag (1 /α, · · · , 1 /α ), α > 0. diagonal: E = diag (1 /α 1 , · · · , 1 /α d ). more complicated: arbitrary E positive definite. heavy tailed: E has eigenvalues λ 1 , · · · , λ d with positive real parts 0 < 1 /α 1 := Re ( λ 1 ) ≤ · · · ≤ 1 /α d := Re ( λ d ) .
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they?
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they? There are some challenges... operator sum-stability: For each n , exists an operator (matrix) A n , such that A n ( X 1 + · · · + X n ) d = X , (2) with X i ’s independent copies of X . In this case, A n = n − E .
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they? There are some challenges... operator sum-stability: For each n , exists an operator (matrix) A n , such that A n ( X 1 + · · · + X n ) d = X , (2) with X i ’s independent copies of X . In this case, A n = n − E . Note that A n commutes with the ‘+’ operation!
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they? There are some challenges... operator sum-stability: For each n , exists an operator (matrix) A n , such that A n ( X 1 + · · · + X n ) d = X , (2) with X i ’s independent copies of X . In this case, A n = n − E . Note that A n commutes with the ‘+’ operation! operator max-stability? For A n = n − E , we cannot always write A n ( X 1 ∨ · · · ∨ X n ) = A n X 1 ∨ · · · ∨ A n X n , since A n may not commute with the ‘ ∨ ’ operation...
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they? There are some challenges... operator sum-stability: For each n , exists an operator (matrix) A n , such that A n ( X 1 + · · · + X n ) d = X , (2) with X i ’s independent copies of X . In this case, A n = n − E . Note that A n commutes with the ‘+’ operation! operator max-stability? For A n = n − E , we cannot always write A n ( X 1 ∨ · · · ∨ X n ) = A n X 1 ∨ · · · ∨ A n X n , since A n may not commute with the ‘ ∨ ’ operation... There is no obvious direct analog of (2) for maxima that covers all stability exponents E ...
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Motivating question and some difficulties Are there operator max-stable distributions and what are they? There are some challenges... operator sum-stability: For each n , exists an operator (matrix) A n , such that A n ( X 1 + · · · + X n ) d = X , (2) with X i ’s independent copies of X . In this case, A n = n − E . Note that A n commutes with the ‘+’ operation! operator max-stability? For A n = n − E , we cannot always write A n ( X 1 ∨ · · · ∨ X n ) = A n X 1 ∨ · · · ∨ A n X n , since A n may not commute with the ‘ ∨ ’ operation... There is no obvious direct analog of (2) for maxima that covers all stability exponents E ... One good approach is via point processes and directional extremes...
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Operator scaling and directional extremes Let X 1 , · · · , X n be i.i.d. with heavy-tailed operator RV( E ) law. Convergence of point clouds under operator scaling: N n := { A n X 1 , · · · , A n X n } ⇒ N , as n → ∞ , where N is a PPP with operator-scaling intensity φ .
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References Operator scaling and directional extremes Let X 1 , · · · , X n be i.i.d. with heavy-tailed operator RV( E ) law. Convergence of point clouds under operator scaling: N n := { A n X 1 , · · · , A n X n } ⇒ N , as n → ∞ , where N is a PPP with operator-scaling intensity φ . (directional extremes) Define the maxima along direction (angle) θ ∈ R d \ { 0 } : n � M n ( θ ) := � X j , θ � . i =1 Note that n n � � M n ( A t � X j , A t n θ ) = n θ � = � A n X j , θ � i =1 i =1 so we expect { M n ( A t n θ ) } θ ∈ R d to have a non-degenerate limit.
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References A Limit Theorem Theorem (Meerschaert, Scheffler, & S.) As n → ∞ , we have { M n ( A t n θ ) } θ ∈ R d = ⇒ { Y ( θ ) } θ ∈ R d , in the space C ( R d ; R ) , where nP { A n X 1 ∈ ·} v → φ ( · ) , n → ∞ .
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References A Limit Theorem Theorem (Meerschaert, Scheffler, & S.) As n → ∞ , we have { M n ( A t n θ ) } θ ∈ R d = ⇒ { Y ( θ ) } θ ∈ R d , in the space C ( R d ; R ) , where nP { A n X 1 ∈ ·} v → φ ( · ) , n → ∞ . Idea of proof: Consider the tail sets B ( θ, r ) = { x ∈ R d : � x , θ � > r } , r > 0 . Then P { M n ( A t n θ ) ≤ r } = P { A n X i ∈ B ( θ, r ) c , 1 ≤ i ≤ n } , which is P {N n ⊂ B ( r , θ ) c } − → P {N ⊂ B ( r , θ ) c } = exp {− φ ( B ( r , θ )) }
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References A Limit Theorem Theorem (Meerschaert, Scheffler, & S.) As n → ∞ , we have { M n ( A t n θ ) } θ ∈ R d = ⇒ { Y ( θ ) } θ ∈ R d , in the space C ( R d ; R ) , where nP { A n X 1 ∈ ·} v → φ ( · ) , n → ∞ . Idea of proof: Consider the tail sets B ( θ, r ) = { x ∈ R d : � x , θ � > r } , r > 0 . Then P { M n ( A t n θ ) ≤ r } = P { A n X i ∈ B ( θ, r ) c , 1 ≤ i ≤ n } , which is P {N n ⊂ B ( r , θ ) c } − → P {N ⊂ B ( r , θ ) c } = exp {− φ ( B ( r , θ )) } = P { Y ( θ ) ≤ r } .
Some Limit Theory Representation and Simulation Testing for Hetero-Ouracity Implementation and Applications References The limit process The limit process Y : is non-negative, max-i.d. with f.d.d. � � m � P { Y ( θ 1 ) ≤ r 1 , · · · , Y ( θ m ) ≤ r m } = exp − φ ( B ( r j , θ j )) , j =1 where B ( r , θ ) := { y ∈ R d : � y , θ � > r } .
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