Cover Times, Sign-dependent Random Walks, and Maxima
Full text
(2) ! "$#%&&' ()#+*,-./#%& 0#21 3# 46575'8"9:8<;>=@?BA757C. D EGF
(3) HJILKMFONQPSROTOTVU W XZY\[O] EGP XZ^ E_FO`7a [ EbK X P [ EbcQHZd e Fgf [ N2h ^ diEbcjEbk$E X F>`%l X HJK ^ FVNQFVm>f D$n%oJpgTVTrqVqsDEGF
(4) HJILKMFONQP Dtun W n'v.
(5)
(6) !#"$&%('*)+, .-0/ # !12.3546
(7) 72/ 8 1 * 93:35,# ;4<2 =3!/ 4>?/ #2@;A1 2AB#64< !,C ,
(8) 354D21,E, FG #1HI 4
(9) JKK F G LM KKN >%( 62,POQ !4<6 1 H#R11,POQ%('1*)S #8
(10) -;/ # !12POTU22V2/ 1,XWYOZ#6[<# M \ Z]_^`Y G 8!W2Ja F 8cb2a1a8cb )ed \ )+@87fg)_8cJ2K1K F 87f.Z87K#J \ ZZ^ G,L K G 8cJJ#Wa \ Zd@^`%=)e-Bh2fg)hi@jK F hK1k8cZl [m @po_4 !'VOZ q 0Hr,4sJKK F n t ,4< u6v6 6 ,46# [v t Z87s]sOrZ#[< wJKK F.
(11)
(12) )> D ! [, c< @23g351'4 qI 1Bq2qr,4< M )>=A4< !4 q2qr,4< 4;[v1 ,R [, [ 6 e (6(35'4 qqI 4>46 26 e ! #RE r 4 #@q41H, M \ ;eA4< cgqqI 4 O246 '6 3514U[v1,4Q g3r qe461 _ g4_ ,4<2 601p354_461 > 2 e ,qr, #e ,#6 [ 2rvCqI1,#6 R !46 H'D
(13) D M )+46 c3514 [< # 'D p246X' ,46 1N. !46 H' M )> 646 !3546 3514
(14) X ! cR 35'I[ 23X[v1,4; N 2qq41q4 6,R [ 2 . = !46 H' & BH62 D M )> c64 H'6 1 (2[v1#1,46 [v=6N !46 H' X246B4<2q [,R ' !4<D M \ p( !D[v1pqqI 4*6B
(15) 9 ,4<2 6;; ! 8 qI ,#>461 >2 M ! 87 qI 1B461 >2 ! q04<21 >2 e,46=0v8 c6,qX6468 q46HH p[ 2 Hr. r 4 1p1 .qI# ! ,# XI23:8? M )> 646 !3546 3514Usq46HI2H cR ,I ! cR g35'I[ E23VA4< !*q 6 62>6 B _ 4 DVO Hr26 354 X 18 ,qr,,# 461 _ e u6$[vI c<2#= q46HH cR 1, 462 ;35'[ 6 1 >2I e qI ,#E 16 [ 2P,CqI1,# 2Rp !46 H'D 6 D Sq[v;4<2I c3514 M Z vC6, 1 B23 A4< !Bq1 621 6 p246 !'I D M )> 4<2I c3514 w23g; ! cRN35'[v X3514@ A4< cBq 6 62(6 ; B [ 2 u c64 H'6 1 1H6 M )> Sq1[v646 c3514 =4' ,46 [ 2R& 8 1,4D p6 c64 H'6 1 @4B ' c6462D HRp146q M )>gA4< cSqI 6 Q * P' ( =U 4<(q2qr,4S@1H6 (646 u6 1=q41H2H u6 D 3514Qs ! 87 qI 1Q461 >2 2 @HR(4<2 !3546 M e3:6,4g !' u<2H 6[, O _ [, 41,[,(354BE 187,qr, #s4<21 >2NNE[v 4!<2 r' . B ! e M \ . qr [v 2([,1 !D 6646 c3514 354 . !46 H' Eqr1 6 H6 ,4 2R# [,R M \ N= 4@[,1 !D ' 4 [,V 46 E >' M )>>35'4 qqI 4 [,[,,46 Q#' 0Hr,4*3V 2C (qr 1< U pB46,,87 ,I ! [,'Hr e 4 qI1 #6 Q246g4<21 RBq[vD M )>* 2 46 'uQ Q2=,C[ QvCq [, u+3546 0' 3514_B246 [vs23P6(#' 0Hr,4>23S 2C M 46, ' U3546 0' *354_6s4 2[,1R [,#6 ;eD B 4 2 R
(16) q26 [ 3514 ;' e [< _ D ( ! 6 M. !#" $ $" % ! & . ). . . ( '. * + ,.-0/124365.798&:&;<5.=9:&:<5.=?>@BAC,@EDC5FGDH@5.8JI&I&;J5FK-L5IM8?>,N-O-L13<P9IL3Q58&897&:?>,N-0A 1$RS@BTVU9FGDH@CUJWCDIM;9U9IMX<5.-LY. .
(17) .
(18)
(19) 46 !U2392 O \ _ '; _6(0 ;R0 !'qI 4 !14 * 43 M "+24< -@ ! M \ 2 s1462v35' 6E ;R[v 21' ss # 123 f., 6 [ 2gZ#62 c6 [ B26
(20) d@R12 \ ! '6 3 )+ [<1R
(21) Z#[<1 M. . Z#[< OZ,q, ;HI 4sJ2K1K F @IEo_246 '.
(22) #.
(23) . d@ v35,46,I[v M= M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M L
(24) !(")*$#%$%&+ '()*%$+,.-0/1+2)'342%*/5 )768/549:6;<(=%?>@3A %? (/54B=.422 C D J MG \ #64'[v MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M W J MJ J MF J ML J Mb. o 1,4_ MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M )> [v 4_
(25) (3514@
(26) H 4878 2Nq4[, 6 M=M=M(M=MBM=M(M=M=M(M=M )>=46([v1,46 E2e M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M )>= u6 p[ (23Q f.246 [<2 e u6N,CqI1,# 2V ;6 D J M b MG )> Sq1[vs4<2 !3546 3514 M=M(M=M=M(M=M J M b M J )>e u6 [, _23r=f.246=[<2 e u6
(27) ,CqI1,# 2
(28) D M=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M J M k o 1[v' 46, 246 M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M d@v35,46,I[v M=M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M. ". FE. <GHJ IKG@LNM,GHOIQP@RS . W. Y. GJ GJ G,L Gb JF JL. TVUW&%$++(+%$X+836;/5<+((+(=N/5C.+%12C.( a`Q)*%?%13A-Y%$Z [%$/5 8_/Z)! +(68<(/549 X+[3/5Z \^]/b\^\^<42_<=a2%$`Q+/Z a`Q%$/b=]3_Z` D FMG \ #64'[v MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M J1W FM J 46 !*q1 621*6 e3514*(4<21 >2 e p 187,qr, #g4<2 u6 1 q41H2H u6 D MBM=M(M=M(M=M=M(M=S$g M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M J1W FM J MG o_2[v'6 1 3Ql;ced@f hji ?k hml M=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M J2Y FM J M J o_2[v'6 1 3Ql;ced kbnpo<q l MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M FG FM J M F )>=q46HH cR ,,4< ;35'[v X23^r M(M=MBM=M(M=M=M(M=M FG FM F 46 !;qI 6 p3514
(29) f.246.[< e u6$ I,qr,,#
(30) ,CqI1,8 6 R !46 H'D ;6 D MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M )>0 23Up 6[,DN[,# ''I @f246 [<2 e u6 ! 87 qI 1 ![<1 c6 [B1
(31) 6 M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M FM b lUC,I ! >3+AI46 !eq 6 s MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M FM b MG )> qI 1@1E6 8 C M=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M FM b M J )> qI 1HI16 8 C M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M FM b M F )>( (3541 = !e ue6pK
(32) '# S2H 46q6 1 M(M=M=M(M=M. . FM L. Fd Fd. . F2L. Fa Lk Lk. La bG.
(33) .
(34) . FM b M L FM b M b. o ,4_6 =23 4<21 M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M )> 23=6X646 c3514 354E, cR 35'I[ 23( [v 4_
(35) M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M qqr, C M=M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M d@v35,46,I[v M=M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M. bG bb b2Y. kG. LI M+0G /5 - \ +#64=%? ) '3[v _.\^ /bMB\M=24M(M=%$M(+FM=-AM=+(M(CM= M(` M=%?MB3AM=%$ M=M(%$M= M(M=/5M( M=M=) M(M=6"MB/54<M=9M(+ M=/Z M=XM(4M=)'M ` k1Tb. . 0 , P !46 H' I MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM J MG 4< c@ !6
(36) !64!6
(37) c< ,46 M=M(M=MBM=M(M=M=M(M=M LIM J M J lgC[v'4< 1 3541 c<6 MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM J M F )>=qI# ! 3:6,4 c6,q e !64!6
(38) X c<6 ,46 M=M(M=M=M(M=M LIM J M L )>=qI# ! 3:6,4 c6,q e !64!6
(39) X
(40) ,,4<2V c<6 M=M(M=M LIM J M b ^@' ,46 [,2V ' !4<6 1 M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM F s !R q62 [, @23Q(646 c3514 M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM L )> #6 6 e2 _ p[v1,46,I[v M=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM LIMG )> 116 6 M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM LIM J &D2E[, 41,[, M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M LIM b ZqID[v 2S[ . O[v q4 _2N' ,46 [,29 ,4< 1 M=MBM=M(M=M=M(M=M LIM b MG ] 46 e 2
(41) e u646 u3:DO !1 2 , 4 M(M=MBM=M(M=M=M(M=M LIM b M J ] 46 e 2
(42) e u62 42 4 3: OI !1 2 ,,46 M=M(M=M LIM b M F ^@' ,46 [,2V 46 1 >2 ' !4< I M=M=M(M=MBM=M(M=M=M(M=M d@v35,46,I[v M=M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M. kk #k W ka WK WG W2J WL WW WW a1b a#W a#W aa aY Y1J. b M J )> fX M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M b M F )>=TU246 [v M=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M b M L Z ;'2 2Xo q4 !1 M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M b M b Z' 4R M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M qqr, CN M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M qqr, CN] M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M d@v35,46,I[v M=M(M=M=MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M. Y1b #Y W. LIM J.
(43) . +. . ((. . 0 0 K. D @.),%$\A+ %?0 - ()*=% /5>#5)*/b/J2/52 &% =/5_%$ %J /b)33%? $+(>2)* + /b2 C 2+<\^a( + -,=% T b MG \ #64'[v MBM=M(M=M(M=M=M(M=M(M=MBM=M=M(M=M(M=M(M=M=M(M=MBM=M(M=M=M(M=M Y1b. GKG G K1b G K#W G1G K G1GG.
(44)
(45) ! _P I AP"! 3$# ! 8%# &'&(&)# !* ./// 32 , P465 // M 87 , P465 !+-, 3 0 // M GG MM qq M3M827 II 353544 P4P + KK /// 1 M GG MM qq M3M82:7<99 353544 PP 66;; KK>= >>)>22 ;e e u 62X&IqIqI 2264<4<22
(46) vv66,,4N4 2 !, 1187N qI ;qr1 11 42 <2 0I12u 3:873:8? > ;2 2]_NHIv < _10, D2X 0 216DE !,;.4B< !_
(47) Eqq46,N 4646I e11 29 M 14 ;.4(? [v XA 4 M )>EqI24<2 v6,4 2 I ;[<1 ,$[ [v146 X66 D ! 46 1461@3P4<2 !35,46,I[v@646'1 sA 4 M )>s41,4 2 I O#( 6 * Bq46HH cR 6e6 187,qr, #>4<2I > 2pq1 >(e H> 24< 6(A,4 M )>
(48) +(@ ><_p 9 ,46,#Hr,I# 4< O;2Hr , 4p 0HI ( , 4IO 35'4 4 14X1 p 46@?D[ AIu6,4EHrv<_, D !X<_ 46, I M )>' E 1, 46 2 23_ ,8?(e , (@A p4(? [v p4<21 > 2 2 I G v OrE461 > 2O 2 R ,
(49) 46,46 [X, 41'O ,, 4 2 I 2 KB= bO =2 , 246H 47<<9 246O R2: 9 )> 7 , 6[v 46v 646
(50) 1 (>I2 2 249 7 7C9 2 I M [,# ''I _21' *q46[, 6 !D M '4!6,46 46O#([,# ''I q4[vD _246( !qr [, [ +23IB[v1# #'' Q21'Use 187,qr, #Q4<2 _ = 1 s 4 Hr,I# 4 M \ _ D * (0]_4(e p 2 e u6 [,,462 EqI24<2 v6,4< *_sqr1 u6 1 <. )> 96 S[v1 !6 93# _>q246 M )>UA46 !Pq4!DOo q 46 PJ L O [v1[v 4I V 2 RB[v1,4 6 D ,O 187,qr, #_4<2 _ *p
(51) 6 >[,D[ 6 e E M o I2q6,4eb= 6 [v1 q24e23Q(6 O e 4B6=' 0Hr,4@3Q 2C
(52) X;46,,87 ,I ! [,'Hr= e !' D M ` 4 4_q'H N1,4< ! 23*o 2q6,4b; o_246 ' JKK F M O 9O 4
(53) . 187,qr, #;4<21 _ 1v8 !,q 4<2I ! e4B HR 354 K G Mq M q 5 3 4 K G M M. . G.
(54) J.
(55)
(56) . (. I23:8? 21 U226,4 ] 46 e 2
(57) 2 1@,12 >u3:87 M )>e46 1 U 46 I [ 6 ;AIu6,4 M )>=[v116 '1' e2I2' 6 !qr [, P[,1 !D e2Hr=246B 6[v' 6 ]_46 2 Z , ?J2KK#J (]_46 e 2. K e u6$46 u3: 46 D[ 6 2 K q21 G J2Y G F K vO9]_46 e 2 26 1 e u6,466 46 u3: q21 G J2a G JY e ]_46 e 2E !1 2 , 4 5q G Ja M 187,qr, #_4<2 _1 5
(58) '4> e 1 2>HI , !R !, 2 [,2R !'I D. & u6,4<'4;HI,3546 M )>0A4< != 'H cD[ (23
(59) A4< c(q4!(23 ;D ! ( 6p,46 (2 vCq46 6 ! E354>6=q46HH M )>(A4< c@ !,qN6p[,2[v'2 6 q41H2H u6 D *1 *Hr,
(60) ,46 s 4<2 !3546 354 A46 ! qI 6 e6 svAD 1 U351 @ M )>( !624 ;qI1 # 3Ps4<2 _
(61) Hrv _, p'qqr,4e2Ip 4 HI2464 ,4SBUA4< c+q 6 6 * V6UAI46 !P U6*4<2 _s# u< +#Rs236 HI2464 ,4< ,O, 4@ 'qqr,4B4e60 _,4 M vCq [, u4<2 !3546 3514@ A4< csq 6 6 =1 >HI , H62 Dp3514,46 #
(62) 4<2 !3546 23QBq41H2H u6 D M )>.646 c3514 N246&[, q [ D M \ j$35 !qr [, =[, E&646 !3546 N246 2R# [,R 4!6 H M - 4 O
(63)
(64) qI# HEvvCq [v
(65) ,Cq4D 1 e 2 2 [,S[v1
(66) q' 4eq414<2 O ef.2q M @3: 4s ' u<2H 6[,2 O6(4<2 !3546 4 2 !s' !D(B,46 U646 !3546 S23 !46 H' M f46, 4 O e u6;[,'q _ 2Q6@ 6[,2
(67) #',I[v>23I@ 18 ,qr,,#U461 _ Q #DO
(68) qR _ [, 41,[,N
(69) Xf246$q4[vD M )>646 !3546 3514p4<2I ! , ucR 23=6 p Nq46[vD E $1,46R [v1 q [,2D M -@ _,1,4DO E 4!6 H u6,4s2R# [,R 5 =[, 4e' ,46 [,2R M =' !
(70) v6Nq41qI# !DE X qI2qr,4eHRE@HI ! G YY#b O e [< RE46v35 4> el*' 4 > v6 M )> !D[v1 !'H c [v A4< c*qI24*23V6@6 *[v1,4g6 D M P, "! ! Hr c6[< ! [=q46[v 6 B2Xv HI q46 , 4 .46 M )>,.[v1,4e6
(71) 23 6
(72) 46 6;A46 !6 ;
(73) q46[v 6 @I s# u6 .E46023U, M i 26 6 [v1,4_ =HR O,. @?. C E 0 ( @A ( E . ( . (. (. (. (. ( !2P4* < O. O (. O. . G. M . G. 5I $ . C @?E. ( !2P B . . " % I. ^E +A I ^ M I ', )3' ! ) -6=. 3$# &% K !'q ! ) ('*)+'. )+ ,@
(74) 14( #' = 3 [,,4_6 OI,e; c6[<I ! [sq4[vD >Hr q <4 2I _ M Pv G D @Hr q[vD N;[v 4<[v'
(75) 35,46,I[v= [, 4<[v0 , 6p4<2 _.c' q$ED ,O ! 1'46 GMGM )> A4< ! 461. ". #. 35476(894. 0 / / / / / / / / / / / / / /0/ / / / / //// // /// / / 1 / 0 0 / / // 2 // // / / // / / / / 0/ / / / 0 / ////////// ///////. :;=< >@?+ACBEDFBED,GIHKJ9LNMOPQRJTSUNJWVJYX[Z]\^MHN_a`cb5MVcbPedgfRHbfR`hOiMVjgJW_ /. >2p)>s1 > c 64 u 6H 'N6 21P U3VD >( [v1, 4g[,6 4< [v'
(76) 35354 4
(77) [vq(B >46 1[, >4 26 U 2M E] 46(e 2 fX8 6 1 @Hr, p #1 ! 12 No 1Oo (> 2 2E-@ !+ JKK1KE M ,4<2 ,>.
(78) F. . 46 '6 3514 s[v 4Q
(79) D g6=;354*= qe4<21 >2 e ,qr,I,#* ,8 6 [ 2U,CqI1,# 2R. c64 H'6 1 N M p2 !X 4 N646 c3514 354( q41H2H ucR , ucR *35'[v 23S6([v 4U6 B354e0 8 ,qr,I,#>46 >29O Hr26 e u6 [, !6#_1
(80) 6 2 I,qr,,#vCqr # 2Rp !46 H' N18 =6 D M '4!6,46 46O _e1e@ !' D4<2 !3546 g2396@ _ c64 H'6 1 M Z = !qr [, S[,1 !D _4= !' D Xv<2 e u6 #'
(81) 4 [,V4D !'u< M )>e1 cUq4!U3I6e6 c6' 6@' ;HI 4U239 C ( pB46,,87 ,I ! [,'Hr M ]_, j35 @ B2,Cq6 123g6vA u6 1.23_p C 0' qr # M \ X _28 , 12 [,1 ! _;' u0 #'246 M \ &6 = #'4 q[,
(82) N' ;HI 4 3sqr #6 ' u3514 R M [< 3s6 qI1 #6 M \ 3sX3X2,4qI1 #< ' u #'4XI pHr26 246 4 8 [v4< 6.2I 41, 4 #8 [v4< 6O [<1 , qr 1 ;. 2C ;' qI1 # M )> ;
(83) D2 0I
(84) 4N246p. 4;qr #
(85) 6'qqr,4=46 1#B46 [v6203546 6qI1 # M '46 G1M J @ =2&vC2 q e u6.64 qr #6 ep6B62V' ;HI 4e23Q 2C ; @J M \ '46 G1M J 5H _4s62R G W qI1 #< e .46,; C M )>
(86) q4 [, q0
(87)
(88) ; 46,
(89) ! I M -@,46 _0I. . . <. . . V. ( ( * 9. &. d. 9. . M. . B B . . . . . d. . . .
(90)
(91) . d. :;=< >@?+A"BED D hS,d J9LMOPNQRJW` d OiM+LfROiMKfRH b S*dC_NfROJWHN`fRdgH` / h UNJ `dgQRfR_ PEdgfRHb`iMVJ OiM+LfROiM MH_ b UNJ7fRVWQRJ7`MVJYHNdb /. . K. ' [v'Hr e,46 ;q1[vE' ;HI 4(23 ' u3514 R c64 H'6 &qI1 #6 M qI1 #= [ 2D
(92) C 0' qI1 # e,N1(23S6=26,4eqr #6 > E6=' [v'HI(1 >41,4 87[,4< I1O#87[,#146 2B287[,#146 2 M \ _28 ! I2>[, E,Cq [v vCq46 6 ! 0354;N4 2[, 6 ,C8 qr [vD ' 0Hr,4_23V C @Hr, p e354_==6 M -@ _, 4 O2s4 2I[v p6B46,,87 ,I ! I[ > R ep'qE;( 0 4 R q8 626 [.vCq46 6 ! M .1H62 D 2 ,Cq [v E3514 ;'3514p6.246 2[,23( ' ;HI 4(23* C E 64
(93) ,I ! M )> s3546 0'E @ OV'
(94) ! [v1,48 1,[,O6>=D
(95) 4 N=1 !R q [BvCq46 6 ! e
(96) D c6 6 3g;246 [v M )>0 c64 H'6 12 [v1,46 [v(3g;#' 0Hr,43U C N 8 ! I sI Bq46, 1' RXHr, eX6NHr46 2 5qr,4< Q[, 0' [ e u6. d. <. ( (. . (. (. (. K. (. .
(97) L.
(98)
(99) . ". '. M is,46R M 01=' !D ;'2 23g6; !46 H' 3Q60' ;HI 4s3U C8 2 [, q4D e Hr26 46 2 !46 H' p [,4, !46 H' 6 ' !4<6B [, 41,[, M. . . . . ". G
(100) G YY#b 9^' ,46 [ 2 \ 1,4< ! (23 +2q[,*)S4<2I c3514 23 41H2H ucRpis c64 H'6 1 M!"$# %'& ,) ( )* +-,PO F k28 L#FM. *. . J .$/10/12345765!98 :;2
(101) 3=< ?J2KK#J $>C @? BA DCE )FcHG
(102) JI K K EL * (F I J2N M ] 46#+2/ ' ,4DO#]>1 ! M. . .. FMON0 :PBQRS JKK F =^2D
(103) 6 U T 4 2I[vp3@ ^@' 0Hr,4
(104) 23sf2C )>4 =i , 1 MJ" , M* # g 5 % ( + O LL K8 LL W M. ( . LTONU/1V!XWY28Z[N\/]U3-^_\'`a/1:P cb JKKK
(105) )> 4' q46H, [v 46 D =23e R
(106) ,46 [
(107) 4<21 _ =&]_46 e 2. MZ% ) d % e (. e T? O G W1WD8 G Y#J M. T. .
(108)
Related documents
In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions.. Paper
Each point in the figures is the mean of 1000 runs and the squares in the grid used to check if the surface is covered or not have the width 10 −4. There is also a 95%
These studies have revealed that under quite general circumstances it is possible to extract a number of characteristic invariants of the dynamical systems without a
In the above example NICETEXT transforms the three-bits of ciphertext into eleven-bytes of innocuous text with a space between words1. The number of letters per word in the
I had worked on the idea of hire purchase for a niche market of foreign students in Swedish universities in the entrepreneurship and business innovation course, I wanted to
Since the study will focus on the remittance corridor between South Africa and Zimbabwe, a remittances sender will in this paper refer to a person who has moved from Zimbabwe to
Extended cover
A Coalescing random walk is an interacting particle system which can be described as the following. Suppose that there is one particle located at each integer position [0, n] at