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|00000000| 58 72 65 66 3a 20 73 70 | 61 72 6b 79 20 73 63 69 |Xref: sp|arky sci|
|00000010| 2e 6d 61 74 68 3a 31 30 | 36 38 36 20 73 63 69 2e |.math:10|686 sci.|
|00000020| 73 79 73 74 65 6d 73 3a | 31 33 36 20 62 69 6f 6e |systems:|136 bion|
|00000030| 65 74 2e 69 6e 66 6f 2d | 74 68 65 6f 72 79 3a 37 |et.info-|theory:7|
|00000040| 38 36 0a 4e 65 77 73 67 | 72 6f 75 70 73 3a 20 73 |86.Newsg|roups: s|
|00000050| 63 69 2e 6d 61 74 68 2c | 73 63 69 2e 73 79 73 74 |ci.math,|sci.syst|
|00000060| 65 6d 73 2c 62 69 6f 6e | 65 74 2e 69 6e 66 6f 2d |ems,bion|et.info-|
|00000070| 74 68 65 6f 72 79 0a 50 | 61 74 68 3a 20 73 70 61 |theory.P|ath: spa|
|00000080| 72 6b 79 21 75 75 6e 65 | 74 21 65 6c 72 6f 79 2e |rky!uune|t!elroy.|
|00000090| 6a 70 6c 2e 6e 61 73 61 | 2e 67 6f 76 21 61 6d 65 |jpl.nasa|.gov!ame|
|000000a0| 73 21 6e 73 69 73 72 76 | 21 6b 6f 6e 67 21 6a 6f |s!nsisrv|!kong!jo|
|000000b0| 73 6c 79 6e 0a 46 72 6f | 6d 3a 20 6a 6f 73 6c 79 |slyn.Fro|m: josly|
|000000c0| 6e 40 6b 6f 6e 67 2e 67 | 73 66 63 2e 6e 61 73 61 |n@kong.g|sfc.nasa|
|000000d0| 2e 67 6f 76 20 28 43 6c | 69 66 66 20 4a 6f 73 6c |.gov (Cl|iff Josl|
|000000e0| 79 6e 29 0a 53 75 62 6a | 65 63 74 3a 20 44 6f 75 |yn).Subj|ect: Dou|
|000000f0| 62 6c 65 20 73 74 6f 63 | 68 61 73 74 69 63 69 74 |ble stoc|hasticit|
|00000100| 79 20 61 6e 64 20 65 6e | 74 72 6f 70 79 20 69 6e |y and en|tropy in|
|00000110| 20 4d 61 72 6b 6f 76 20 | 70 72 6f 63 65 73 73 65 | Markov |processe|
|00000120| 73 0a 4d 65 73 73 61 67 | 65 2d 49 44 3a 20 3c 31 |s.Messag|e-ID: <1|
|00000130| 39 39 32 41 75 67 32 39 | 2e 30 32 35 39 31 30 2e |992Aug29|.025910.|
|00000140| 36 37 33 31 40 6b 6f 6e | 67 2e 67 73 66 63 2e 6e |6731@kon|g.gsfc.n|
|00000150| 61 73 61 2e 67 6f 76 3e | 0a 53 65 6e 64 65 72 3a |asa.gov>|.Sender:|
|00000160| 20 6a 6f 73 6c 79 6e 40 | 6b 6f 6e 67 2e 67 73 66 | joslyn@|kong.gsf|
|00000170| 63 2e 6e 61 73 61 2e 67 | 6f 76 20 28 43 6c 69 66 |c.nasa.g|ov (Clif|
|00000180| 66 20 4a 6f 73 6c 79 6e | 29 0a 4f 72 67 61 6e 69 |f Joslyn|).Organi|
|00000190| 7a 61 74 69 6f 6e 3a 20 | 47 6f 64 64 61 72 64 20 |zation: |Goddard |
|000001a0| 53 70 61 63 65 20 46 6c | 69 67 68 74 20 43 65 6e |Space Fl|ight Cen|
|000001b0| 74 65 72 0a 44 61 74 65 | 3a 20 53 61 74 2c 20 32 |ter.Date|: Sat, 2|
|000001c0| 39 20 41 75 67 20 39 32 | 20 30 32 3a 35 39 3a 31 |9 Aug 92| 02:59:1|
|000001d0| 30 20 47 4d 54 0a 4c 69 | 6e 65 73 3a 20 34 30 0a |0 GMT.Li|nes: 40.|
|000001e0| 0a 5c 64 6f 63 75 6d 65 | 6e 74 73 74 79 6c 65 5b |.\docume|ntstyle[|
|000001f0| 31 31 70 74 5d 7b 61 72 | 74 69 63 6c 65 7d 0a 0a |11pt]{ar|ticle}..|
|00000200| 5c 62 65 67 69 6e 7b 64 | 6f 63 75 6d 65 6e 74 7d |\begin{d|ocument}|
|00000210| 0a 0a 49 20 68 61 76 65 | 20 61 20 72 61 74 68 65 |..I have| a rathe|
|00000220| 72 20 73 69 6d 70 6c 65 | 20 28 49 20 74 68 69 6e |r simple| (I thin|
|00000230| 6b 29 20 69 6e 66 6f 72 | 6d 61 74 69 6f 6e 20 74 |k) infor|mation t|
|00000240| 68 65 6f 72 79 20 70 72 | 6f 62 6c 65 6d 20 77 68 |heory pr|oblem wh|
|00000250| 69 63 68 20 49 20 63 61 | 6e 27 74 0a 73 65 65 6d |ich I ca|n't.seem|
|00000260| 20 74 6f 20 66 69 6e 64 | 20 61 64 64 72 65 73 73 | to find| address|
|00000270| 65 64 20 69 6e 20 6d 79 | 20 74 65 78 74 73 20 28 |ed in my| texts (|
|00000280| 41 73 68 2c 20 7b 5c 69 | 74 20 49 6e 66 6f 72 6d |Ash, {\i|t Inform|
|00000290| 61 74 69 6f 6e 20 54 68 | 65 6f 72 79 7d 3b 20 4d |ation Th|eory}; M|
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|000002b0| 64 2c 20 7b 5c 69 74 20 | 4d 61 74 68 65 6d 61 74 |d, {\it |Mathemat|
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|00000310| 49 20 6d 61 79 20 6e 65 | 65 64 20 69 73 20 61 20 |I may ne|ed is a |
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|00000360| 69 6e 69 74 65 20 75 6e | 69 76 65 72 73 65 20 24 |inite un|iverse $|
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|000003d0| 5f 69 0a 5c 68 62 6f 78 | 7b 20 61 74 20 74 69 6d |_i.\hbox|{ at tim|
|000003e0| 65 20 7d 20 74 20 29 2c | 20 74 20 5c 69 6e 20 49 |e } t ),| t \in I|
|000003f0| 24 20 61 6e 64 20 74 72 | 61 6e 73 69 74 69 6f 6e |$ and tr|ansition|
|00000400| 20 6d 61 74 72 69 78 20 | 24 52 20 3d 20 5b 20 72 | matrix |$R = [ r|
|00000410| 5f 7b 69 6a 7d 20 5d 2c | 0a 72 5f 7b 69 6a 7d 20 |_{ij} ],|.r_{ij} |
|00000420| 3d 20 70 28 20 78 5f 69 | 20 5c 68 62 6f 78 7b 20 |= p( x_i| \hbox{ |
|00000430| 61 74 20 74 69 6d 65 20 | 7d 20 74 2b 31 20 7c 20 |at time |} t+1 | |
|00000440| 78 5f 6a 20 5c 68 62 6f | 78 7b 20 61 74 20 74 69 |x_j \hbo|x{ at ti|
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|00000470| 5e 74 5f 69 20 3d 20 5c | 73 75 6d 5f 6a 20 70 5e |^t_i = \|sum_j p^|
|00000480| 7b 74 2d 31 7d 5f 6a 20 | 72 5f 7b 69 6a 7d 24 2c |{t-1}_j |r_{ij}$,|
|00000490| 20 61 6e 64 20 65 6e 74 | 72 6f 70 79 20 24 48 28 | and ent|ropy $H(|
|000004a0| 20 70 5e 74 20 29 20 3d | 0a 5c 73 75 6d 5f 69 20 | p^t ) =|.\sum_i |
|000004b0| 70 5e 74 5f 69 20 5c 6c | 6f 67 5f 32 28 20 70 5e |p^t_i \l|og_2( p^|
|000004c0| 74 5f 69 20 29 24 2e 20 | 0a 0a 49 74 27 73 20 75 |t_i )$. |..It's u|
|000004d0| 73 75 61 6c 20 74 6f 20 | 64 65 66 69 6e 65 20 74 |sual to |define t|
|000004e0| 68 65 20 75 6e 63 65 72 | 74 61 69 6e 74 79 20 6f |he uncer|tainty o|
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|00000500| 20 70 72 6f 63 65 73 73 | 20 61 73 20 74 68 65 20 | process| as the |
|00000510| 6c 69 6d 69 74 0a 6f 66 | 20 74 68 65 20 63 6f 6e |limit.of| the con|
|00000520| 64 69 74 69 6f 6e 61 6c | 20 65 6e 74 72 6f 70 79 |ditional| entropy|
|00000530| 20 24 5c 6c 69 6d 5f 7b | 74 20 5c 72 69 67 68 74 | $\lim_{|t \right|
|00000540| 61 72 72 6f 77 20 5c 69 | 6e 66 74 79 7d 20 48 28 |arrow \i|nfty} H(|
|00000550| 20 70 5e 74 20 7c 20 70 | 5e 7b 74 2d 31 7d 2c 0a | p^t | p|^{t-1},.|
|00000560| 70 5e 7b 74 2d 32 7d 2c | 20 5c 6c 64 6f 74 73 2c |p^{t-2},| \ldots,|
|00000570| 20 70 5e 30 20 29 24 2e | 20 0a 0a 42 75 74 20 77 | p^0 )$.| ..But w|
|00000580| 68 61 74 20 49 27 6d 20 | 69 6e 74 65 72 65 73 74 |hat I'm |interest|
|00000590| 65 64 20 69 6e 20 69 73 | 20 74 68 65 20 6c 6f 6e |ed in is| the lon|
|000005a0| 67 2d 72 75 6e 20 62 65 | 68 61 76 69 6f 72 20 6f |g-run be|havior o|
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|000005c0| 20 74 68 65 0a 73 74 61 | 74 65 20 76 65 63 74 6f | the.sta|te vecto|
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|000005e0| 65 6c 66 2c 20 74 68 61 | 74 20 69 73 20 24 5c 6c |elf, tha|t is $\l|
|000005f0| 69 6d 5f 7b 74 20 5c 72 | 69 67 68 74 61 72 72 6f |im_{t \r|ightarro|
|00000600| 77 20 5c 69 6e 66 74 79 | 7d 20 48 28 20 70 5e 74 |w \infty|} H( p^t|
|00000610| 0a 29 24 2e 20 49 6e 20 | 70 61 72 74 69 63 75 6c |.)$. In |particul|
|00000620| 61 72 2c 20 61 20 63 6f | 6c 6c 65 61 67 75 65 20 |ar, a co|lleague |
|00000630| 6f 66 20 6d 69 6e 65 20 | 68 61 73 20 73 75 67 67 |of mine |has sugg|
|00000640| 65 73 74 65 64 20 74 68 | 61 74 20 24 48 28 20 70 |ested th|at $H( p|
|00000650| 5e 74 20 29 24 20 67 72 | 6f 77 73 0a 6d 6f 6e 6f |^t )$ gr|ows.mono|
|00000660| 74 6f 6e 69 63 61 6c 6c | 79 20 69 66 20 61 6e 64 |tonicall|y if and|
|00000670| 20 6f 6e 6c 79 20 69 66 | 20 24 52 24 20 69 73 20 | only if| $R$ is |
|00000680| 60 60 64 6f 75 62 6c 79 | 20 73 74 6f 63 68 61 73 |``doubly| stochas|
|00000690| 74 69 63 27 27 2c 20 74 | 68 61 74 20 69 73 20 24 |tic'', t|hat is $|
|000006a0| 5c 66 6f 72 61 6c 6c 0a | 69 2c 20 5c 73 75 6d 5f |\forall.|i, \sum_|
|000006b0| 6a 20 72 5f 7b 69 6a 7d | 20 3d 20 31 24 2c 20 6e |j r_{ij}| = 1$, n|
|000006c0| 6f 74 20 6a 75 73 74 20 | 24 5c 66 6f 72 61 6c 6c |ot just |$\forall|
|000006d0| 20 6a 2c 20 5c 73 75 6d | 5f 69 20 72 5f 7b 69 6a | j, \sum|_i r_{ij|
|000006e0| 7d 20 3d 20 31 24 2e 0a | 0a 41 6e 79 20 68 65 6c |} = 1$..|.Any hel|
|000006f0| 70 20 77 6f 75 6c 64 20 | 62 65 20 61 70 70 72 65 |p would |be appre|
|00000700| 63 69 61 74 65 64 2e 20 | 50 6f 73 74 73 20 61 72 |ciated. |Posts ar|
|00000710| 65 20 66 69 6e 65 2c 20 | 6f 74 68 65 72 77 69 73 |e fine, |otherwis|
|00000720| 65 20 49 27 6c 6c 20 70 | 6f 73 74 20 74 68 65 0a |e I'll p|ost the.|
|00000730| 65 6d 61 69 6c 20 73 75 | 6d 6d 61 72 79 2e 0a 0a |email su|mmary...|
|00000740| 54 68 61 6e 6b 73 20 69 | 6e 20 61 64 76 61 6e 63 |Thanks i|n advanc|
|00000750| 65 2e 0a 0a 5c 65 6e 64 | 7b 64 6f 63 75 6d 65 6e |e...\end|{documen|
|00000760| 74 7d 0a 0a 4f 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |t}..O---|--------|
|00000770| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000780| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000790| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000007a0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000007b0| 2d 2d 3e 0a 7c 20 43 6c | 69 66 66 20 4a 6f 73 6c |-->.| Cl|iff Josl|
|000007c0| 79 6e 2c 20 43 79 62 65 | 72 6e 65 74 69 63 69 61 |yn, Cybe|rneticia|
|000007d0| 6e 20 61 74 20 4c 61 72 | 67 65 2c 20 33 32 37 20 |n at Lar|ge, 327 |
|000007e0| 53 70 72 69 6e 67 20 53 | 74 20 23 32 20 50 6f 72 |Spring S|t #2 Por|
|000007f0| 74 6c 61 6e 64 20 4d 45 | 20 30 34 31 30 32 20 55 |tland ME| 04102 U|
|00000800| 53 41 0a 7c 20 53 79 73 | 74 65 6d 73 20 53 63 69 |SA.| Sys|tems Sci|
|00000810| 65 6e 63 65 2c 20 53 55 | 4e 59 20 42 69 6e 67 68 |ence, SU|NY Bingh|
|00000820| 61 6d 74 6f 6e 20 20 20 | 20 20 20 4e 41 53 41 20 |amton   |   NASA |
|00000830| 47 6f 64 64 61 72 64 20 | 53 70 61 63 65 20 46 6c |Goddard |Space Fl|
|00000840| 69 67 68 74 20 43 65 6e | 74 65 72 0a 7c 20 63 6a |ight Cen|ter.| cj|
|00000850| 6f 73 6c 79 6e 40 62 69 | 6e 67 73 75 6e 73 2e 63 |oslyn@bi|ngsuns.c|
|00000860| 63 2e 62 69 6e 67 68 61 | 6d 74 6f 6e 2e 65 64 75 |c.bingha|mton.edu|
|00000870| 20 20 20 20 6a 6f 73 6c | 79 6e 40 6b 6f 6e 67 2e |    josl|yn@kong.|
|00000880| 67 73 66 63 2e 6e 61 73 | 61 2e 67 6f 76 0a 56 20 |gsfc.nas|a.gov.V |
|00000890| 41 6c 6c 20 74 68 65 20 | 77 6f 72 6c 64 20 69 73 |All the |world is|
|000008a0| 20 62 69 73 63 75 69 74 | 20 73 68 61 70 65 64 2e | biscuit| shaped.|
|000008b0| 20 2e 20 2e 0a          |                         | . ..   |        |
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