~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~ ~ ~ ~ ~~~ ~~~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~~~~~~~~~~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~ ~~ ~~~ ~~~~~ ~~~ ~ ~ ~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~ ~~ ~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~~~~~ ~~~ ~~ ~ ~~~~~~ ~~~~ ~~ ~ ~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~ ~~~~~~~~ ~~ ~~ ~~~ ~~~~~ ~~~~~~~~ ~~~~~~~~ ~~~~~~~ ~~~~ ~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~ ~ ~~~~~~~~~~~~~~ ~~ ~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~ ~~~~~~~~~~~~~~ ~~~~~ ~~ ~~~~~~~~~~ ~ ~ ~~~~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~ ~ ~~~~~~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~ ~ ~~~~~~~ ~ ~~~~~~~~~ ~~~~~~~~~~ ~ ~~~~~~ ~ ~~~~~~~~~ ~ ~~~~~~~~ ~~ ~~~~ ~~~~~~~~ ~ ~ ~ ~~ ~ ~ ~~~~~~ ~ ~~~~~~~~~ ~ ~~~ ~~~~ ~~~ ~ ~ ~ ~~ ~ ~~~~~~ ~~~~~~~~ ~ ~ ~ ~~ ~ ~~~~~~~ ~~~~~ ~ ~ ~ ~~~ ~ ~ ~~~~~~~~~~ ~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~ ~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~ ~ ~~~~~~~ ~ ~~~~~~~~~~~~ ~ ~~~~ ~ ~~~ ~~ ~~~ ~ ~ ~~ ~~~~~~~~~~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~ ~ ~~~~~~ ~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~ ~~~ ~~~ ~ ~ ~~ ~~~~ ~ ~ ~~~~~~~~~~ ~~ ~~ ~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~ ~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~ ~~~~~~~ ~ ~ ~~~~~ ~~ ~~ ~ ~ ~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~ ~ ~~~~~~~~~ ~~~ ~~~~ ~~~~ ~~~~~~~~ ~~~~~~~ ~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~ ~ ~~ ~ ~~~ ~~~~~~~~~ ~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~ ~~~~~~~ ~ ~~~ ~~ ~~~ ~~ ~~~ ~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~~~~~ ~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~ ~~ ~~ ~~~ ~~~ ~~ ~ ~ ~~ ~~ ~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~ ~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~ ~~ ~~ ~~~~ ~~~~~~~ ~~ ~ ~~ ~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~ ~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~ ~~~ ~ ~ ~~~ ~~ ~~~ ~~~~~ ~~~~~~~~~~~~~~~~ ~ ~~~ ~~ ~~~~~~~~~~~~~~ ~~~~~~ ~~ ~ ~ ~ ~~~~~~ ~~~~~~~~~~~ ~~ ~ ~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~ ~~~~ ~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~ ~~~~~~~~~~ ~~~ ~~ ~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~ ~~ ~ ~ ~ ~~~~~ ~~~~~~~~ ~~~ ~~~~~~~~~ ~~ ~ ~ ~~~ ~~~~~~~ ~~~ ~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~ ~ ~ ~~~~~~ ~ ~ ~ ~ ~ ~ ~~ ~ ~~~~~~~~~ ~ ~ ~~~~~ ~~ ~ ~~ ~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~ ~ ~ ~~~~ ~~~ ~ ~~ ~ ~~~~~~~ ~~~ ~~~~~~~ ~~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~~ ~ ~~~ ~~ ~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~ ~~~~~~~~~~~~~~ ~~~~~~~ ~~~~ ~~~ ~~~~~ ~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~ ~~ ~ ~ ~~ ~~~~~~ ~ ~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~ ~~~~~~~~~~~~~~ ~~~~ ~ ~ ~~~ ~~ ~~~ ~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~ ~~~ ~~~ ~ ~~~~ ~ ~~~ ~~~ ~~~~~~ ~~~ ~~~~ ~~ ~~~~ ~~~~ ~~ ~~~~ ~~~ ~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~ ~ ~ ~~ ~ ~~ ~ ~~~ ~~~ ~ ~~ ~~ ~~~ ~~~~ ~~~ ~ ~~~~~~~~~~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~~~~~~~~ ~~ ~~ ~~ ~~ ~~ ~~~ ~~~ ~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~ ~~~ ~~~~~~ ~~~~~~~~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~ ~ ~~~~~~~~~~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~~~~~~~ ~~~~~~ ~~~ ~ ~ ~~~~ ~~~ ~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~ ~~ ~ ~ ~ ~~~~ ~ ~ ~~~ ~~ ~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~ ~ ~ ~~~~ ~ ~~ ~ ~~ ~~ ~~~~~ ~ ~~~~~~ ~ ~~~~~~~~~~~~ ~ ~ ~~ ~~~ ~ ~~~~~~~ ~~~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~ ~~~~~~~~ ~~ ~~~~ ~ ~ ~~~~ ~ ~ ~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~ ~ ~~~~~~~~~~ ~~ ~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~~ ~~~ ~~ ~ ~~~ ~ ~ ~~~~~ ~~~ ~~ ~ ~~~ ~ ~ ~~~ ~~ ~~~ ~~~~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~ ~~~ ~~ ~~~ ~~~~~ ~~~~~~~~~~ ~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~ ~ ~ ~ ~ ~~~~ ~~~ ~ ~ ~~~~~~~~ ~~~~~~ ~ ~~ ~~~ ~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~ ~ ~~~~~~~~ ~ ~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~ ~~ ~~~ ~~~~~~~~ ~ ~~~~~~ ~ ~~~~~~ ~ ~ ~ ~~~~ ~ ~ ~~~ ~~ ~~~ ~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~ ~ ~~~~~~ ~ ~~~~~~ ~~~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~~~~~~~~~ ~~~ ~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~~~~~~~~~~~~~~ ~~ ~ ~~ ~~~~ ~ ~~~~~~~~~~~~~ ~ ~~ ~ ~~ ~ ~ ~ ~ ~~~~~~~~~~~~ ~~~~~~ ~ ~ ~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~ ~ ~~~~~~~~~~~~~ ~~ ~~~~~~~ ~~ ~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~~ ~~~~~ ~ ~~~~ ~ ~~~~ ~~~~~~ ~~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~~ ~~~~~~~ ~~~~~~ ~~~~~~~~~~ ~~~~~ ~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~~~~ ~ ~~~~~~~~~~ ~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~ ~~~~~ ~ ~~~~~~~~ ~~~~ ~~ ~ ~ ~~~ ~~ ~~~ ~~~~~ ~ ~ ~~~~~~~~ ~ ~~ ~ ~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~ ~~~ ~~~~~~~~ ~~~~~~~~~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~ ~ ~~~~~~ ~ ~~~~~ ~~~~~ ~~ ~~ ~ ~~~~~~~~~~~ ~~~~~~~~~~~~ ~~~~~~ ~~~ ~ ~~~~~~~ ~ ~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~ ~~~ ~~~ ~~ ~~~ ~~ ~~~~ ~~~~~~~~~~ ~~~~~~~~~~~ ~ ~ ~ ~ ~ ~~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~ ~ ~~~ ~~ ~~~ ~~~ ~~ ~~~~ ~~~~~~~ ~ ~~~~ ~ ~~~ ~ ~~ ~~~ ~ ~~~~~~~~ ~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~~ ~~ ~ ~ ~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~ ~~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~ ~~ ~~~~~ ~ ~~ ~ ~~~~ ~~~~~~~~~~~~ ~ ~~~ ~ ~~ ~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~ ~ ~~~~~~~~~~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~~~~~~ ~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~ ~ ~ ~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~ ~~ ~ ~ ~~ ~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~ ~~~~~~~ ~~~ ~ ~~~ ~~ ~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~~~~ ~ ~ ~~~ ~~~~ ~ ~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~ ~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~ ~~~~~ ~~~~ ~ ~~~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~~~~ ~~~~ ~ ~~~ ~~~~ ~~~~~~~~~ ~~~~ ~ ~ ~~~~~~~~~~~~~~~ ~ ~~ ~~~ ~~ ~ ~ ~ ~ ~ ~ ~~~~ ~~~ ~ ~~~~~ ~~ ~~ ~~~~~~~~~~ ~ ~~~~~~ ~ ~ ~ ~~~~~~~ ~~~ ~~~~~ ~ ~~ ~ ~ ~~~ ~ ~ ~ ~~~~~ ~ ~~~ ~ ~~~ ~~ ~ ~ ~~~ ~ ~ ~ ~~ ~ ~~~~~ ~ ~~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~ ~~ ~~~~~~ ~ ~ ~~~ ~~~~~~~~~ ~~ ~~~~~ ~~~~ ~ ~~ ~ ~~~~~ ~ ~ ~ ~~~ ~ ~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~ ~ ~~~~~~~ ~~~~~~~~ ~ ~~~~~~ ~~ ~ ~~~~~~ ~~~~~~~~~~ ~ ~ ~~~~~~~~~~ ~ ~~~~~~~~ ~ ~~ ~ ~~ ~~~~~~~~~ ~~ ~ ~ ~~~~ ~ ~ ~~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~ ~ ~ ~ ~~ ~~ ~~~~ ~ ~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~ ~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~~ ~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~~~~ ~~ ~ ~~ ~ ~~~~ ~~ ~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~~ ~~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~ ~~ ~ ~ ~ ~~ ~~~~~~ ~~~ ~~ ~ ~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~~~~ ~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~ ~ ~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~ ~ ~ ~~ ~ ~ ~ ~~~~~~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~~~ ~~ ~~~ ~ ~ ~~~~ ~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~ [...]... 2n ( 1- p 2 )2 x2 - 2 + y2 dxdy 27r + 17 3 WEAK DEPENDENCE FOR GAUSSIAN SEQUENCES ° ~ 1 e -4 x 2 dx =P(A)+2n-1 J 21p exp 2 2 x P - Jo + 313 2(1-p 2) P XY y2 2(1-p2) 1-p2 - - e - Y I 2 dy J It follows easily that P(AB)-P(A)P(B) > {exp [3/2(1-p2) ]-1 -Icjl}P(A), where cj - 0 as j-* co, which contradicts the uniform mixing condition The investigation of Gaussian sequences satisfying the strong mixing... form (17 1 3), where the ~, are independent and normally distributed Corollary 17. 3.2 The condition (17 1 7) is necessary and sufficient for the regularity of a Gaussian sequence with spectral density f (;,) Theorem 17 3 2 A Gaussian sequence X; satisfies the uniform mixing condition of and only if the a-algebras 9JIk ~, sufficiently large n k+n are independent for all Proof The sufficiency of the... If (2)-T(2)I E If the order of T (A) is then for N, k > N, 7C f e` 1k T ( 22)d2 = 0 n Hence, for n > N, rz p(n) = sup P,Q f ein~P(2)Q(A)f(a)d2 = sup f P,Q J _ -rc - e`n)P(2)Q(2)(f(2)-T (2))d2 7r rz E sup P,Q f I P ( 2 ) I I Q (2 ) I d2 I n < Em -1 sup P,Q Em -1 f -n sup I Q = Em - 1 < -n P (2) I I Q (2) I I f (A)I& IP(2)1 2 f(2)d2 J 7r -7 r - n IQ(2)I 2 f( ;2)d~ The continuity off (2) has of course... is non-zero for infinitely many values of j Without loss of generality, suppose that E (X;) = 0, E (X;) = 1 Let j satisfy Ri 0 0, and write Rj = p Define events B={XXE[0,1]} A={X0>2/p}, If Xj is uniformly mixing, then P (AB) - P (A) P (B) I where 0 U) < O(j)P(A), as j-+ oc But *0, P (AB) - P (A) P (B) = oo f2 I /P f0 - f exp - "' I exp P 2/ _ 0 x2 -2 pxy+y2 dxdy 2(1 - p 2 ) 2n ( 1- p 2 )2 x2 - 2 +... matrix R defines the distribution of a random vector with characteristic function (17 3.1) The following properties are immediate consequences of the definition (1) The variables X1 , X2 , , Xn are independent if and only if R kj =O for k 0j, i e if and only if they are uncorrelated (2) If Xj = (XI j , X2j, , Xnj), and if the vector (X 1 , X2 , Xm) is Gaussian, then Y-, bj Xj is Gaussian for all... the matrix (R tj, t ) is symmetric and positivesemi-definite In particular, if (Xj) is a stationary Gaussian sequence, then any condition of weak dependence of the a-algebras SJJ1k J , Jnn+k can, in principle, be expressed in terms of the autocovariance function of the sequence, or of the spectral function Such expression may be far from simple, and raises difficult and interesting analytical problems... representation of the form (17 1 3) In this representation the variables C; are limits in mean square of linear combinations of the X X , so that the vector (~ I , , ~„) is Gaussian Since the ~; are orthogonal, they are independent From (17. 1 3), H' (X) c H' ( ), H_~,(X) c H_,,, ( ), and since ~ ; is regular by the zero-one law, Xj must be Corollary 17. 3 1 The Gaussian sequence Xj is regular if and only... = 314 WEAK DEPENDENCE FOR STATIONARY PROCESSES Y j > „bj Xj - with E I Y1 2 = E IZI 2 = 1 rz sup p (fl) = f e on, P ( ;) Q Chap 17 Hence, from the results of © 16 6, ( 22)f (2) d2 , -n where the supremum is taken over all trigonometric polynomials P and Q of the form aj e"' , P b k e`"k , Q (17. 3 2) k >-0 j-> 0 with n 7r f IP(2)1 2 f(2)d2 IQ ( )12f (4')& 2 =f 7r -7 r By the Weierstrass approximation... 17 3 WEAK DEPENDENCE FOR GAUSSIAN SEQUENCES 311 It is easy to see that aj = E(Xj), Rkj = E {(Xk -ak)(Xj - aj)} If the matrix R is non-singular, then X has the probability density (where IRI is the determinant of R) , p(x1, x2, , xn ) = (2 t) Z" IRI ? exp {- LErkj(xk -ak)( xj - aj)} , where the matrix (rkj) is the inverse of R Conversely, any positive-semidefinite matrix R... away from the theme of this book, and we shall discuss them only in order to construct examples of processes satisfying the conditions of ©© 1, 2 Theorem 17. 3 1 A Gaussian sequence Xj is regular if and only if it is linearly regular 3 12 WEAK DEPENDENCE FOR STATIONARY PROCESSES Chap 17 Proof It suffices to show that every linearly regular Gaussian sequence is regular By Lemma 17 1 1 such a sequence