lecture8 - 5&4� vivde 3 who{he dujoifif cmmnanfl dams...

Unformatted text preview: 5&4» { vivde 3 who {he dujoifif cmmnanfl dams That is, Far-MW g So that? ' s = C. U C; U ‘ ' U Ck - CS fl Cf = 95 3' 5 1- t - I? h I e C 5 , finer: (6" j ' ( 6 C5,, J 6 Ct; S "f t: Him efifhem' i '*’ j’ av J ‘h' C Star 2" Causiour wold CLUS Safarwwj, and Mamivve whether I? is and, recuwmé, {unsung etc. 30, fix a dds; C; aw}: (a) Start by attacking .12 +41: class C i§ closed w not (Her) can be dune using M5 4“: graft», wiHaouf looking at {he fwbabflfln‘as «M W ulan dais (‘95 Naif duck :? 41M- .‘5 armbabz 0v Shawn M Udm‘s can also be 4m direcfly «GM 4w graph) (C) IF 4m class Cgsiinilve, ma: (‘5‘) IF if is closed, if Can be .1415 class is PosiHve rewweut (Hm is an “mum o! M Em- Frubuius 6:604:91) . we restw‘cf fke Hui-her, it Markov Chain “1'0 use mlj ~H~e Mmifimc be tween Shh! M C , 4142 Vesuan chain E! irreducible M ’30: Hi ve raw-"wt M a 1Qth 5h!“ srace,‘ for such chums fiwc exisk (MI; C K “of 'u Music/«If- (‘Dfl J‘he clan C is mi? C I'8 mi dosed, (MI? C rs mused chum reskvxz'ted b [Nutrient dawwfi'on, basil—We {plum-#- clued. «Men ‘4‘ Es I am "H49 C Wu We a MIT/ll invavimd' 3’ («MR , 4km ’ Mnsc'euf- Markov cm :3 (60 Mam/fie, C L! clnxd, W? me $86 ham:th analysis 15 deMe 4' C .3 Maxim («f whim 1mth fw a 5M: m: i5 (us my. 1) or i! C 5: MM recuwml: (if Mum rem"! HM; fir a 5M2 (5C {.5 infinif-g). [Kean ‘- cVMj claw v23} be nouwmt N Wsiwf. W H- fS rccuwwl‘, {1‘ mg be. fan‘ch rccuwwf 0/ mu rum/mt. .ngtzf 3: Find all fusibie invarimt my;wa 95% «HM chain. Fm“: on Mama (a!) F?" each rm‘h've recuch closed dag C, We eat-3h a an we invariant thWm i1: {'01 «Fae Mark” phth resMcfed 40 (L. (b) Any luvan f‘ ahbufim fix M Mrleh maul Mar/Luv ohm! :3 abhr'ned by sow imam)! WHMHM a! Hue individual IQ '5 find in ruff CA)- LLSter 4 '- Dem/“2w: JAM Immuj bekaytw 4 4w Marla" qu- (5’ 4m: Shay, Mum-1 all alum are 0L w: av Sunny v.11 wan {a (.9) IF J‘ is {raujiwh Ham (3‘) =0. (.0) 1}" )' is null reoqu 4W (4‘) =0. [Noi' w ens +0 show, M 464 s M.) 7 “27 (d) 50 mm alsum [39f and J‘ is fpflfi‘va recurrent Let— C be ML ammumcafinfl 61w (’70:er wwwwk and dual) M (mm; j. 9m Smfz, we know W M Marlow dun/i (“Law 40 C i; Ewduafic, FAII'HVZ newt/uh M a win, so #13 rufan om Ln; a WW7“; 063%}un ’Tf’c' war C, M {A atria“!- sz P(x..=} Ixek) '~ Iraq) V NC. “~90 So not)! let T be «Hue 33f «Hm fie chain W73 C ; we km: 1 P(Xn =j 'xo‘() 3 T4“, Xa‘i) ‘F(TSnIX.=i) = z ?(x.=J' l xT=k,T< n, X~=2> rec . = < =') T709 HT-«,X. c 'P(T§ n H. =e) B” 4‘“! Malta! "ml 1%) X1 = k, 4‘“? WM T- atum/A behavu Bike 0! Marker chain darling M k. 5:. '- 33-: m. =jl¥7=k, Tsn, x.=2) = 113(5). HUM: a) : gt T<w, Xoci) 2 Ami“ 1;." W12» 1 ¥o=i) = P(T<~IX.=E) rah Combinan 44M“ «fact, W Anal: P(T<00’ Xa=i) «new 0‘ WM :1 em“ EM“: 2m '1 a J' h (H = w! s 4» it’d“ 17c (j) W 3 rake-b m 171' F a Sh? 5'. 17e4'erwfiwz ions {wt averaju- w 17mm m4 LOX 10 be a bowm Vevmrck fan‘ch W M gm brace? flew: emsv}; a Cowfiwfir V1 8: H”(:)I< M qaamux We gum. MRr€5+€J “:1 : N-«I . Z M79) M (H) H-iao N 3:? (K0 5 {‘ 0M F [S M a fasiéwk reCawu‘IL Cmmum‘cabkj (1055' C, W 3 967?) = Z «M 43v) ( J‘GC c) J where fiat k 414 (“Von/funk diS‘Fflbuhm h m mm «’54va 41, C. ((> I“? 79 '(r and t is I}! & Wmsiewf COMWMVUCMT-Mg Claw; C, “Kw 44w Jim/“Ur 075%: IS 0M3 J'JmM‘jHr’fi/WWA owe/r ¥ . Lat {CM 172 HM W? H2 «U Fos'rH/e recurrent WMMMUAHV‘j Clauses . $19 we asswul‘ 2: {WM cu yaw) = CL res-{hive recwwd' (*7?) = (L regihh 2: 5L [PU/1G CQ(X,H‘)[2: «C 9946)} .396» k (ecu MAL ...
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