Exact asymptotics for linear processes Magda Peligrad University of Cincinnati October 2011 (Institute) October 2011 1 / 27
Exact asymptotics for linear processes Plan of talk -Early results -Central limit theorem for linear processes -Functional central limit theorem for linear processes -Selfnormalized CLT -Exact asymptotic for linear processes (Institute) October 2011 2 / 27
Early results: i.i.d. …nite second moment Theorem Let ( ξ j ) be i.i.d., centered at expectation with …nite second moments. ∑ n j = 1 ξ j p n ! σ N ( 0 , 1 ) and [ nt ] j = 1 ξ j ∑ p n ! σ W ( t ) Here σ = stdev ( ξ 0 ) . (Institute) October 2011 3 / 27
CLT for linear processes with …nite second moments ∞ n ∑ ∑ X k = a k + j ξ j , S n = X j , j = � ∞ j = 1 Theorem (Ibragimov and Linnik, 1971) Let ( ξ j ) be i.i.d. centered with …nite second moment, ∑ ∞ k = � ∞ a 2 k < ∞ and σ 2 n = var ( S n ) ! ∞ . Then D S n / σ n ! N ( 0 , 1 ) . ∞ σ 2 b 2 ∑ n = nj , b n , j = a j + 1 + ... + a j + n . j = � ∞ It was conjectured that a similar result might hold without the assumption of …nite second moment. (Institute) October 2011 4 / 27
Functional central limit theorem question for linear processes. For 0 � t � 1 de…ne [ nt ] i = 1 X i W n ( t ) = ∑ σ n where [ x ] is the integer part of x . Problem Let ( ξ j ) be i.i.d. centered with …nite second moment, ∑ ∞ k = � ∞ a 2 k < ∞ and σ 2 n = nh ( n ) with h ( x ) a function slowly varying at ∞ . ( h ( tx ) / h ( x ) ! 1 for all t as x ! ∞ ) . Is it true that W n ( t ) ) W ( t ) , where W ( t ) is the standard Brownian motion? This will necessarily imply in particular that for every ε � 0, 1 � i � n j X i j � εσ n ) ! 0 as n ! ∞ . P ( max (Institute) October 2011 5 / 27
Functional CLT. Counterexample. Example There is a linear process ( X k ) such that σ 2 n = nh ( n ) and such that the weak invariance principle does not hold: 1 P ( j ξ 0 j > x ) � , x 2 log 3 / 2 x 1 1 1 a 0 = 0 , a 1 = log 2 and a n = log ( n + 1 ) � log n , for n � 2 , n � n / ( log n ) 2 and lim sup σ 2 n ! ∞ P ( max 1 � i � n j ξ i j � εσ n ) = 1 . (Institute) October 2011 6 / 27
Functional CLT. When E ( j ξ 0 j 2 + δ ) < ∞ and σ 2 n = nh ( n ) then functional CLT holds. W n ( t ) ! W ( t ) with W n ( t ) standard Brownian motion Merlevède-P (2006). When E ( ξ 2 0 ) < ∞ and σ 2 n = n λ h ( n ) with λ > 1 then W n ( t ) converges weakly to the fractional Brownian motion W H with Hurst index λ / 2. Fractional Brownian motion with Hurst index λ / 2, i.e. is a Gaussian process with covariance 2 ( t λ + s λ � ( t � s ) λ ) for 0 � s < t � 1. structure 1 (Institute) October 2011 7 / 27
CLT for i.i.d. centered with in…nite second moments H ( x ) = E ( ξ 2 0 I ( j ξ 0 j � x )) is a slowly varying function at ∞ . ( � ) De…ne b = inf f x � 1 : H ( x ) > 0 g � s : s � b + 1 , H ( s ) / s 2 � j � 1 � η j = inf , j = 1 , 2 , � � � Theorem Then ∑ n j = 1 ξ j p nH n ! N ( 0 , 1 ) and [ nt ] j = 1 ξ j ∑ p nH n ! W ( t ) where H n = H ( η j ) (Institute) October 2011 8 / 27
Selfnormalized CLT for i.i.d. centered with in…nite second moments Giné, Götze and Mason(1997) Theorem H ( x ) = E ( ξ 2 0 I ( j ξ 0 j � x )) is a slowly varying function at ∞ is equivalent to ∑ n j = 1 ξ j q ! N ( 0 , 1 ) ∑ n j = 1 ξ 2 j and [ nt ] ∑ j = 1 ξ j q ! W ( t ) ∑ n j = 1 ξ 2 j where H n = H ( η j ) (Institute) October 2011 9 / 27
CLT for linear processes with in…nite second moments X 0 = ∑ ∞ j = � ∞ a j ξ j is well de…ned if ∑ a 2 j H ( j a j j � 1 ) < ∞ , j 2 Z , a j 6 = 0 Theorem (P-Sang, 2011) Let ( ξ k ) k 2 Z be i.i.d., centered. Then the following statements are equivalent: (1) ξ 0 is in the domain of attraction of the normal law (i.e. satis…es ( � ) ) (2) For any sequence of constants ( a n ) n 2 Z as above and ∑ ∞ j = � ∞ b 2 nj ! ∞ the CLT holds. ( i.e. there are constants D n such that S n / D n ! N ( 0 , 1 )) . (Institute) October 2011 10 / 27
Regular weights and in…nite variance (long memory). a n = n � α L ( n ) , where 1 / 2 < α < 1 , E ( ξ 2 0 I ( j ξ 0 j � x )) = H ( x ) Example Fractionally integrated processes. For 0 < d < 1 / 2 de…ne Γ ( i + d ) X k = ( 1 � B ) � d ξ k = ∑ a i ξ k � i where a i = Γ ( d ) Γ ( i + 1 ) i � 0 and B is the backward shift operator, B ε k = ε k � 1 . For any real x , lim n ! ∞ Γ ( n + x ) / n x Γ ( n ) = 1 and so n ! ∞ a n / n d � 1 = 1 / Γ ( d ) . lim (Institute) October 2011 11 / 27
Regularly varying weights and in…nite variance; normalizers. De…ne b = inf f x � 1 : H ( x ) > 0 g � s : s � b + 1 , H ( s ) / s 2 � j � 1 � η j = inf , j = 1 , 2 , � � � n : = c α H n n 3 � 2 α L 2 ( n ) with H n = H ( η n ) B 2 where Z ∞ 0 [ x 1 � α � max ( x � 1 , 0 ) 1 � α ] 2 dx g / ( 1 � α ) 2 . c α = f (Institute) October 2011 12 / 27
Invariance principle for regular weights and in…nite variance (long memory). a n = n � α L ( n ) , where 1 / 2 < α < 1 , n � 1 , E ( ξ 2 0 I ( j ξ 0 j � x )) = H ( x ) , L ( n ) and H ( x ) are both slowly varying at ∞ . Theorem (P-Sang 2011) De…ne W n ( t ) = S [ nt ] / B n . Then, W n ( t ) converges weakly to the fractional Brownian motion W H with Hurst index 3 / 2 � α , ( 1 / 2 < α < 1 ). Fractional Brownian motion with Hurst index 3 / 2 � 2 α , i.e. is a Gaussian 2 ( t 3 � 2 α + s 3 � 2 α � ( t � s ) 3 � 2 α ) for process with covariance structure 1 0 � s < t � 1. (Institute) October 2011 13 / 27
Selfnormalized invariance principle Theorem (P-Sang 2011) Under the same conditions we have n 1 ! A 2 where A 2 = ∑ P X 2 a 2 ∑ i i nH n i = 1 i and therefore p c α S [ nt ] q ) A W H ( t ) . ∑ n i = 1 X 2 na n i In particular S n ) N ( 0 , c α q A 2 ) . ∑ n i = 1 X 2 na n i (Institute) October 2011 14 / 27
Higher moments. Exact asymptotics. We aim to …nd a function N n ( x ) such that, as n ! ∞ , P ( S n � x σ n ) = 1 + o ( 1 ) , with σ 2 n = k S n k 2 2 . N n ( x ) where x = x n � 1 ( Typically x n ! ∞ ) . We call P ( S n � x n σ n ) the probability of moderate or large deviation probabilities depending on the speed of x n ! ∞ . (Institute) October 2011 15 / 27
Exact asymptotics versus logarithmic Exact approximation is more accurate and holds under less restrictive moment conditions than the logarithmic version log P ( S n � x σ n ) = 1 + o ( 1 ) . log N n ( x ) For example, suppose P ( S n � x σ n ) = 10 � 4 and N n ( x ) = 10 � 5 ; then their logarithmic ratio is 0 . 8, which does not appear to be very di¤erent from 1, while the ratio for the exact version is as big as 10. (Institute) October 2011 16 / 27
Nagaev Result for i.i.d. Theorem (Nagaev, 1979) Let ( ξ i ) be i.i.d. with P ( ξ 0 � x ) = h ( x ) x t ( 1 + o ( 1 )) as x ! ∞ for some t > 2 , and for some p > 2 , ξ 0 has absolute moment of order p. Then n ∑ P ( ξ i � x σ n ) = ( 1 � Φ ( x ))( 1 + o ( 1 )) + n P ( ξ 0 � x σ n )( 1 + o ( 1 )) i = 1 for n ! ∞ and x � 1 . (Institute) October 2011 17 / 27
Nagaev Result for i.i.d. Notice that in this case N n ( x ) = ( 1 � Φ ( x )) + n P ( ξ 0 � x σ n ) . If 1 � Φ ( x ) = o [ n P ( ξ 0 � x σ n )] then in we can also choose N n ( x ) = 1 � Φ ( x ) . If n P ( ξ 0 � x σ n ) = o ( 1 � Φ ( x )) we have N n ( x ) = n P ( ξ 0 � x σ n ) . The critical value of x is about x c = ( 2 log n ) 1 / 2 . (Institute) October 2011 18 / 27
Linear Processes. Moderate and large deviation Let ( ξ i ) be i.i.d. with P ( ξ 0 � x ) = h ( x ) x t ( 1 + o ( 1 )) as x ! ∞ for some t > 2 , and for some p > 2, ξ 0 has absolute moment of order p . Theorem (P-Sang-Zhong-Wu, 2011) Let S n = ∑ n i = 1 X i where X i is a linear process . Then, as n ! ∞ , ∞ ∑ P ( b n , i ξ 0 � x σ n ) + ( 1 � Φ ( x ))( 1 + o ( 1 )) P ( S n � x σ n ) = ( 1 + o ( 1 )) i = � ∞ holds for all x > 0 when σ n ! ∞ , ∑ ∞ k = � ∞ a 2 k < ∞ and b nj > 0 , b n , j = a j + 1 + � � � + a j + n . (Institute) October 2011 19 / 27
Zones of moderate and large deviations De…ne the Lyapunov’s proportion B nt where B nt = ∑ D nt = B � t / 2 b t ni . n 2 i nt ) 1 / 2 with a > 2 1 / 2 we have For x � a ( ln D � 1 k n ∑ P ( c ni ξ 0 � x σ n ) as n ! ∞ . P ( S n � x σ n ) = ( 1 + o ( 1 )) i = 1 nt ) 1 / 2 with b < 2 1 / 2 , we have On the other hand, if 0 < x � b ( ln D � 1 P ( S n � x σ n ) = ( 1 � Φ ( x ))( 1 + o ( 1 )) as n ! ∞ . (Institute) October 2011 20 / 27
Application Value at risk (VaR) and expected shortfall (ES) are equivalent to quantiles and tail conditional expectations. Under the assumption lim x ! ∞ h ( x ) ! h 0 > 0 P ( S n � x σ n ) = ( 1 + o ( 1 )) h 0 x t D nt + ( 1 � Φ ( x ))( 1 + o ( 1 )) . Given α 2 ( 0 , 1 ) , let q α , n be de…ned by P ( S n � q α , n ) = α . q α , n can be approximated by x α σ n where x = x α is the solution to the equation h 0 x t D nt + ( 1 � Φ ( x )) = α . (Institute) October 2011 21 / 27
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