Independent And Stationary Sequences Of Random Variables - Chapter 9 ppsx

~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~ ~~~ ~~~~~~~~~ ~ ~~~ ~ ~~~ ~~~~~~~~ ~ ~ ~~~ ~ ~ ~~~ ~ ~ ~ ~~~~~~ ~ ~~~ ~ ~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~ ~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~~~~~ ~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~ ~ ~~~~~~ ~~~~~~ ~~ ~~~~~~~ ~~~~~~~ ~ ~~~ ~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~ ~ ~ ~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~ ~ ~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~ ~ ~~~~~ ~~~~~~~~ ~ ~ ~~~~~~~~~ ~~ ~ ~~~ ~~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~ ~~~~~~~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~ ~~~ ~~~~~~ ~ ~~~~~~~~~~ ~~~~~~~~ ~ ~~~~~~~~~~~~~~~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~ ~ ~ ~ ~ ~~~~ ~~~~ ~ ~~~~~~~ ~~~ ~ ~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~ ~ ~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~ ~~ ~~~~~~~~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~ ~~~~~~~~~~ ~~ ~ ~~~~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~ ~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~ ~~~~~ ~ ~~~~~ ~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~~ ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~~~~~~~~~~ ~~~~ ~ ~~~~ ~~ ~~~~~ ~~~~~~~~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~ ~~~~ ~ ~~~~~~~~ ~~~~~ ~~ ~~~~ ~ ~~ ~~ ~~~ ~~~~~~~ ~ ~ ~~~~~~~~~~~~ ~~ ~~ ~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~ ~~~~ ~~~~ ~~~~~~~~ ~~ ~ ~~~~~~~ ~ ~ ~~~ ~ ~ ~ ~~~~~ ~~~~~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~ ~ ~~~~ ~~~~~~ ~~~ ~~~~~ ~ ~~ ~ ~~~~~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~ ~ ~~~~~~~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~ ~ ~ ~~ ~ ~~~~~~~ ~~~ ~ ~~~ ~ ~~~ ~ ~ ~ ~ ~~~ ~~~~~~~~ ~ ~ ~ ~~~ ~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~~~ ~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~ ~ ~ ~~~ ~ ~~~ ~ ~ ~~ ~ ~~ ~ ~ ~~ ~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~~~ ~~~~ ~~~~~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~ ~~ ~~ ~~~ ~ ~~~~~~~~~~~ ~~ ~~ ~~~ ~~ ~~~~~~~ ~~~~~~~~~ ~ ~~~~~~~ ~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~ ~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~~~~ ~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~ ~~~ ~~ ~~~~~~~~~~~~ ~ ~~ ~ ~~~~~~~~~~~~~~ ~~ ~~~~ ~ ~ ~ ~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~ ~~ ~~~~ ~ ~ ~ ~ ~~ ~~~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~ ~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~~~~~~ ~~~~~~~~~~~~~~ ~~ ~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~ ~~~~~~~~~~ ~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~ ~~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~~ ~~ ~~ ~~~ ~~~~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~ ~ ~~~~~~~~~~ ~ ~ ~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~~~~~~~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~ ~~~ ~~~~~~~~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~ ~~~~ ~ ~~~ ~~~ ~ ~ ~ ~ ~~~ ~~~~~~~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~ ~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~ ~~ ~~ ~~~~~ ~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~ ~~~~~~ ~~ ~~~~~~~~~~~~ ~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~ ~~~~ ~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~ ~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~ ~ ~ ~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~ ~ ~ ~~~~~ ~ ~~~ ~ ~~~ ~~~~~~~~ ~~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~ ~~~ ~~ ~~ ~~~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ ~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~ ~~ ~~ ~ ~ ~~ ~~~~ ~ ~ ~~ ~~ ~~~ ~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~ ~~ ~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~ ~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~~~~~ ~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~ ~~~~ ~~ ~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~ ~ ~~~ ~ ~ ~~~ ~ ~ ~~~~~ ~ ~~~ ~ ~~~ ~~ ~~~ ~ ~~~~ ~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~ ~~~ ~ ~~~~~~~ ~~~~~~~~~~ ~~ ~ ~ ~ ~~~~~~~~~~~ ~~ ~~~ ~ ~~~~~ ~~ ~~ ~~~ ~ ~ ~~~~~~~~~~~~~~~ ~~~~~~~~~~~~ ~ ~~~~~ ~ ~~~~~~~~~~~~~~~~ ~~~ ~ ~ ~~ ~~~~~~~~~ ~ ~ ~~ ~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~ ~~ ~ ~~~ ~~~~~~~~ ~ ~~~~ ~~~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~ ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~ ~ ~ ~ ~ ~~ ~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~~~~~~~~ ~~ ~ ~~ ~~ ~~~ ~~ ~~ ~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~ ~~~ ~ ~ ~ ~ ~~~~ ~~~ ~~ ~~~~~~~~~~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~ ~ ~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~ ~~ ~~~~~ ~~~~~~~~~ ~ ~~~~~~~~~~~ ~ ~~~~~~~ ~ ~ ~ ~ ~ ~~ ~~~ ~~ ~ ~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~~~~ ~~~ ~ ~~ ~~ ~~~~ ~~ ~~~ ~~~~~~~~~~ ~~ ~~ ~~~ ~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~ ~~~~~~~~ ~~~~ ~~~ ~~~~ ~~~~ ~~ ~ ~ ~~~~~~~~~ ~ ~~ ~~~~~~ ~~~~ ~~~~~ ~~~ ~ ~ ~ ~~~ ~ ~~ ~~ ~ ~~ ~ ~ ~~~ ~ ~ ~ ~~~ ~ ~ ~ ~ ~~ ~~~ ~~~~~ ~ ~ ~ ~~~~ ~~~ ~~~ ~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~~ ~~~~ ~ ~~~~~~~~ ~~~~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~ ~~~~~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~~~~~~~~~~ ~~~~~~~~~~~ ~~ ~ ~ ~~ ~~ ~~~~ ~~~~~~~~~~~ ~ ~ ~ ~~~~~~~ ~ ~~ ~~ ~~~~ ~~~ ~~ ~ ~~~~ ~~~ ~~~~ ~ ~~ ~~ ~~~~ ~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~ ~~~~~ ~ ~ ~ ~~~~~~~~~~ ~~~~~~~~~~~~ ~ ~~~~~~ ~ ~ ~ ~~~ ~ ~ ~ ~ ~~ ~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~~~~~~~ ~ ~~~~~ ~~ ~~~~~~~~~~~~~~ ~ ~~ ~~~ ~ ~~~~~~ ~ ~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~ ~ ~~ ~~ ~~~ ~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~~ ~~ ~~~ ~~ ~~~~~~ ~~~ ~~~ ~~~~~~~~~~~~ ~ ~~ ~~ ~~~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~ ~ ~ ~ ~~~~~~~~~~~ ~~ ~~ ~~~~~~~~ ~~ ~ ~ ~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~~ ~ ~~~~~~~~~~~ ~ ~~ ~~ ~~~ ~~ ~~~ ~~ ~ ~~~~ ~~~~~~~~~~~~~~~~ ~~~~ ~ ~ ~~~~~~~ ~ ~~~~~~~~~~ ~~ ~~ ~~~ ~~~ ~~ ~~ ~~~~~ ~ ~ ~~~ ~ ~ ~~~~~ ~ ~ ~~~ ~~~ ~~~ ~ ~~ ~~ ~~~~~~~~ ~ ~~~~~~ ~ ~ ~ ~~~ ~ ~ ~ ~ ~~ ~~~ ~~ ~~ ~ ~ ~~~~~~~~ ~~ ~~ ~~~~~ ~ ~~~~~~~~ ~ ~~~~~~~~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~~~ ~~~~~~ ~ ~~~~ ~ ~~~~ ~ ~ ~ ~~~~~ ~ ~ ~ ~~~ ~~ ~ ~ ~ ~ ~ ~ ~~~ ~ ~ ~~ ~~ ~~~ ~ ~ ~~~~~~~~~~~ ~ ~ ~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~ ~ ~ ~~~ ~~~ ~~ ~~~~~~~~~ ~~~~ ~~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~ ~~~~ ~ ~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~ ~ ~ ~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~ ~~~~~~~~~ ~~~~~~~ ~ ~ ~~ ~ ~~ ~ ~~~ ~~ ~~~~ ~ ~~ ~ ~ ~~~ ~ ~ ~ ~~ ~ ~~~~~~ ~ ~ ~ ~~~~~~~ ~~ ~~~~ ~~~~ ~ ~~~~ ~~~ ~~~ ~~~ ~~~~ ~~~~~ ~ ~~~~~~ ~ ~~ ~~ ~~~ ~~ ~~ ~~~~ ~~~~~~ ~~~ ~~~~~~~ ~ ~~~~~~~ ~~~ ~~~ ~ ~~ ~~ ~~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~~ ~~~~~~~ ~~~ ~ ~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~ ~ ~~ ~ ~ ~~ ~ ~~ ~~ ~~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~ ~~~ ~~ ~~ ~~~ ~~~~~~~~~~ ~~~~~~~~~ ~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ ~~~~~~~ ~ ~~~ ~~~~~ ~ ~~ ~~~~~~~~~~ ~~~ ~ ~~~~~ ~~~~ ~~~ ~~~ ~~~~~~~~~~~ ~~~ ~~~~~~ ~ ~ ~~~~~ ~~~~ ~~ ~~~ ~~~~ ~ ~~~ ~~ ~ ~~~ ~~~~~~~~~~~~ ~ ~ ~~~ ~~~ ~~~ ~~~ ~ ~~~~~~ ~ ~ ~~~ ~~~ ~~~~ ~~ ~ ~~~~~ ~ ~ ~ ~ ~ ~~ ~~~~ ~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~~~ ~~~~~~~~ ~~ ~~ ~~~~~ ~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~ ~ ~ ~~ ~~~ ~ ~~ ~~ ~~~~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~ ~~ ~ ~~ ~ ~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ ~~ ~ ~~~ ~ ~ ~ ~ ~~~~~~ ~ ~ ~ ~ ~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~ ~ ~ ~ ~ ~~~~~~~~~~ ~ ~~ ~ ~~ ~ ~ ~~~~~~~~ ~ ~ ~ ~~~ ~ ~ ~~ ~~ ~~~ ~~~~~~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~~~ ~ ~~~ ~~~~~ ~~ ~~~ ~~ ~ ~~ ~~ ~~~~ ~~~~~~~~~~~~~ ~~~~~~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~ ~ ~ ~~~ ~~ ~~ ~~~~ ~ ~ ~~~ ~ ~ ~ ~~ ~ ~~~~~~~~~ ~~~~~ ~ ~~ ~ ~~~~~~ ~~~~~~~ ~ ~~ ~~~~ ~ ~~ ~~~ ~ ~~~ ~ ~~~~~~ ~ ~~ ~~ ~~~~ ~~~~~~~ ~ ~~~~~~~~~~~ ~~~ ~~ ~~ ~~~ ~ ~~ ~~ ~~ ~~~~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~ ~~~~~~~ ~~~ ~ ~~ ~~~ ~ ~~~ ~~~~~~~~~~~ ~ ~~~~~ ~~~ ~~~~~~ ~ ~ ~~~ ~~~~ ~~~~ ~ ~ ~~~~~~~~~~~~~ ~~ ~~ ~~~~ ~~ ~~ ~~~~ ~~~ ~~ ~~ ~~~~~ ~~~~ ~~~~~ ~~ ~~~~~~~~~ ~ ~~~~~~~~ ~ ~~~ ~~~~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~ ~~~ ~ ~~~ ~ ~~~~~ ~ ~ ~~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~~~~ ~ ~ ~~~~~~~~~~~ ~~ ~~ ~~~~ ~~ ~ ~~ ~~~~~~~~~ ~ ~ ~ ~ ~~ ~~~ ~~~~ ~~ ~~ ~~~~ ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~ ~ ~ ~~~~~ ~~~ ~~~~~~ ~~~~ ~ ~~~ ~~~~ ~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~ ~~~~~ ~ ~~~ ~ ~~ ~~~ ~~~~~~~ ~ ~~~~~~ ~~~~~~~~ ~~~~~~~ ~~~ ~ ~~ ~~ ~~~ ~~~~~~~ ~~~~~ ~ ~ ~~~~~~ ~ ~ ~ ~ ~~~ ~~ ~~ ~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~~~~ ~~~ ~ ~~~ ~~ ~~~~ ~~ ~~ ~~~~ ~~~ ~~ ~~ ~~~~ ~~~~~~~~~ ~~ ~~ ~~~ ~~~~~~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~ ~ ~~ ~~ ~~~ ~~~~~~~~ ~ ~~~~ ~~~~~ ~~~~~~~ ~ ~ ~~~ ~ ~~~ ~~~ ~ ~~ ~ ~ ~ ~~~~ ~ ~~ ~~ ~ ~~~~~~ ~ ~~ ~~~~~~~~~~~~~ ~~~ ~ ~~ ~~ ~~~~ ~~~~~~ ~~~~~ ~~ ~~ ~~~~ ~~~ ~~ ~~ ~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~ ~~ ~~~~~~~~ ~~ ~~ ~~~~ ~~~~~~~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~ ~~~~~~ ~ ~ ~ ~~~ ~ ~ ~ ~~ ~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~ ~~ ~~~~~ ~~~~~ ~ ~ ~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~~~~ ~ ~~ ~ ~~~ ~ ~ ~ ~~ ~~~~ ~~~~~~~ ~~ ~ ~ ~ ~ ~~~~~~ ~ ~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~ ~~~~~ ~~~~ ~~~~~~~ ~~~~~~~~~ ~~~ ~~ ~~~ ~~~~~~~~~~~~~ ~ ~ ~~ ~~ ~~~~ ~ ~ ~ ~~~~~ ~~~~~~~~~~~~ ~ ~~~ ~ ~~~ ~ ~~ ~~~~ ~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~ ~~ ~~ ~~~ ~~~~ ~ ~ ~~~ ~~~~~~~~~~~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~~~ ~ ~ ~~~ ~ ~ ~ ~ ~~~~~ ~ ~~~ ~~ ~~~ ~ ~ ~~~ ~~~ ~~ ~ ~ ~~~~ ~~~ ~~ ~~ ~~ ~~~~~~~~ ~~ ~~~ ~~~ ~ ~~ ~ ~ ~~~ ~ ~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~ ~~~ ~~ ~~ ~~~~~ ~ ~ ~~~ ~~~~ ~ ~ ~~ ~ ~~~ ~~~~~~~~ ~ ~~~ ~ ~~ ~~ ~~~~~~~~~~~ ~ ~~~~~~~ ~~ ~~ ~~~~ ~~~~~ ~ ~ ~~~ ~~~ ~~~ ~ ~~~~ ~ ~~~~~ ~~~~~~ ~ ~ ~ ~~~~ ~ ~~ ~~ ~~~~ ~~~ ~~ ~~~~~ ~ ~~~ ~ ~ ~~~~~~~~ ~ ~ ~ ~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~~~~~~~~~~ ~ ~ ~~ ~ ~~~ ~ ~ ~ ~ ~~~ ~ ~~~~~ ~ ~~~ ~~ ~ ~~ ~ ~ ~~ ~ ~~~~~ ~~ ~ ~~~~~~~~~ ~ ~ ~ ~~~ ~ ~ ~~ ~~~ ~~~~~~~ ~~~ ~ ~~~~ ~~ ~~ ~~~~ [...]... [q(B+ 1-2 p){2-p l log n+log c} + -p log q-( 1-2 p) log q+log q-p log p+ -( 1-2 p) log ( 1-2 p)] = =B exp [q(B+ 1-2 p){2-pl) log n+log C1 +p log q] Multiplying this by n- -! q = exp (-2 q log n) , and by Xq q! (9 6.5) (9 6.6) = B exp (pl q log n) , we obtain the expression B exp[q(B+( 1-2 p)(2-p 1 ) log n-(2-p 1 ) log n+ +p log q+log C( 1-2 p))] = = B exp [q(B+log c( 1-2 p)-p log K 1 )] (9 6.7) If p < 4, Ilog C ( 1- 2p)... 9 6 COMPLETION OF THE PROOF 1 87 00 _ e -1 nin'/z - °, r 2 n r! ~rd~ = r 0 0 r n r!n1r J nin e/Z - e = Br exp (-+ i ° n1-Z°1) j, ( _1~2 i r) and rd n -cZ - 1)r ( 9 5 20 ) B'F(2r)n-~i-°1)r = B exp (Br+2r log r-r(z -, u 1 ) log n) _ = B exp (Br +r(a l log n-2 log K 1 -cc, log n)) _ = B exp (Br- 2r log K1 ) = =Be -c5r (9. 5.21) Therefore, summing (9 5 20) over r < m, we get an... 2«1) ; B exp (- 11712 n (9. 5 22) a similar argument obtains for the integral over (-co, -r~ln =- 1) Thus (9 5 16) may be written 'n J_" e - z ~2 1 + r-I ring e -` xd Zr ~r n (x) = e +R 2 , 27c (9. 5 23) 3 where R 2 satisfies the same equation (9 5.17) as R1 P „ 6 Completion of the proof In view of (9. 5 23), we need to study the integrals n-Zr 00 a-z e e - it rd~ = n +rHr§)(x)e 2x2 , -0 0 (9 6 1) where... and the Hr (r< m) are the Hermite polynomials [164] C -1 r] Hr (x) = r ! Z S=0 (2x)q - 2s ( -1 )s s ! (q - 2s) ! ' (9 6.2) We suppose that 0