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Markov Chains and Stochastic Stability

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- +d < /  // - ^ k ` p 9 + 3- 5 @  `Ka g  B< F „
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+ 1 / p - `ba g B< 1 - A 5 / / V  /  + 1 - g / p - A + 1 / +   9 = 9 = !+, !+
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+ <- g ( <  9  j5 1D5 \ 1 - cGA 1 - + < 5 \ /  / / -83=-4 < 2/ 2- +  9 = 9 = +0 +
9 = : - s j./‚B<  ‡ s -43
<- e Z + 1 + + 3 +   - 1D+ V + 1 : : 9 l " ; l F f+ 1 g  BV 21 
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d  5  @
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c TCdTe(fe]TWRbLH d H'b (< R_E


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d + .- 901 eTB<  5 3@./ 1 @ZVA/ + < + '-.- 
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S 
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+ ‚e -4<- 1 5 2/ \
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d -73 +0+ Z=/  B<  <?-*a g   <  O +5B52+d F
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=<B<?/ -LA A  5db \  <=3?3=-‡6<-
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f 1D+9 / @  <63 # g@+5  .` - + 1 - 9 2/b +5B52+Dd 0/  / 
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9 + 1 -?/ \   g 5  Z - DC  + 10+  f - 5 5 - 1 g V B < F

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g V B < / - + 1 \  + 1 90 <B\^\ - 1 F ` - d + 1 +  f+ eTA- 5        9 @ + <  {   {    {{    
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3-$/   5  V1 / g A 5@1D5 \_ZG\ p ^
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d - /  €   - a 1'+ + B/ + < FonGFonGF ` - @g + <=<- g / + <uZ-$/ d -$-4< `ba g V B<  <=3 / - - s @./.-4< g - + 
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+ < 5 \ /  /|A  h  < + <a - 9> /r\ B < /.- 1  + 1 j 5   - 5 \ / + 90   - / -  5  g / + < +  / -
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+    +g   /.-43  1 -  B <  n{   <=3[B <  + eZA 1 <  m  F
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+ <ae + B <e(B < /.- 1 -./ F
f B <=3VB <Ue @g 1 2 /.- 1    + 1[g  B <V / + C - @g + < /B <GA2 /r\ =  1D+ - 1 /2 -X@  9 + 3=- 5 aZB\Ba 9 + 3- 5
- s - 1 g @.- ZV= A /,/ - 1 - A 5 / + < 5 B <-   1  <=3 < + < 5  <-   1 \G./.- 9 - 1 -  1 ->B <A/.-4<63-43 / + eTAB 3-
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1 +
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+ 9 - 3-$/   5 F
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db 5  / +  €
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a
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@g +9>Vg / "# A '3 + 1 ' e 1'+ A   d#  g X1 -Y`Ka g  B <VK  <=3 + < 5 \ˆ / -YB < @g 1 - 9 -4<A/ 9 -  A 1 -
 8 +9 - @g + < C +5 = A / + <   +Dd - 1 < + <a  <UeTA 5@  1 d 2 / 1 -  - g / / + 9 1 e /= "#  179 -  A 1 - F ` -.-
1 -A 5 / C -z Z -$-4< - s /.-4<63-43 / + 1  <63 +9 db 5  + <[.- 9  ae 1D+ A  bZG\ " +   e <  #B<   F
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A - @g + <=32 / + < 9  mT=8@  `ba g B <(@
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ZV A /0/ - =  1'+ +  - 1 -Y@. +9 - d# / 9 + 1 - / 1 < V  1 -4< / F
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 ! #" $


† <  5 \G <Ue / - 1 -A 5 /  <=3 g@+ < g -  / +  1 / † B<>/ - 3 +90 B<V + )/ 9 -#.- 1 2- + 1 \ a
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3- @g 1   / + < +  / - db \0B< d#  g < + < 5  <- 1)9 + 3- 5 9( \>Z-  <  5 \  -43 d 2/ B< / - #|1 + a
C   < @g + < /.- s / 3- C - 52+ -43 / A  1 F S - d  5B5kg@+ <B3- 1 Z + / / -_e -4<- 1  5 < + < 5 B<- 1 '/  /.-
Vg - 9 + 3=- 5   <=3[ +9 -Y  - g ‡ g 65  g / + < d  g /  - + <>/ @ V1 / g A 5@1  + 1D9 F
` - V /./.- 1 < + ?  / @  <  5 \Gj_j / + g@+ < B3- 1 ‡ 1 ./X +9 - 1 / g A 5@1 ./ 1 A g /A 1  5 + 1
./  ZV 5 2 / \   - g / +  / -   +g   /.-43 3-$/.- 1D9  <@./ gug@+ < / 1'+5  + 1 # 9 = 9 + 3- 5‚ <=3 / -4<
A <=3=- 1 6
=  1'+=
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^ a
3-4< 2 / \ @g + <632 / + <  8B< / - 5 B <-  1 9 + 3=- 5 +  - g / + < F  Fon =/ + C - 1  \  1 - 5@ /.-43u./ 1 A g /A 1  5
+ 1 '/  ZV 5 2 /r\   - g / +   / -J'/ +Gg ./ g < + < 5 B <-  1 ./  /.- Vg - 9 = 9 + 3=- 5 F  ^^
" e =5 @e / +  / @03A  5 2 / \  1 -
% e'&R / -   +g   /.-43 # 9 = 9 + 3- 5 @
g Y"Q S"Q h S6T T -4R$<>/  U- R  9  ^ ^  + 9 1D9 =
$, +  @g + < / 1'+5B5@ Z 5 2/r\U=
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9 ^ - g / + < F@{ = )

% eNe'&   + 1D9 +   1'1 -43A g ZV 5 2/ \ 9 / 


- - s @'/.-4< g - +    I U Y mS U@U +S ]]Q S T^]&0\ u;]?S ]WR
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9 + 3=- 5 =P@ / -4< -cGA C 5 -4< / / + / -  G + g   /.-43  ^^ = 9 + 3- 5 Z-B<Ue  h
a  1'1 4- 3VA g Z 5 -

9
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% eNeNe'& / - - s @./.-4< g - +   - 1  + 3V g g$5@ .-K + 1 / -! + 1 d€1 3 gg - Z 5 - # 9 = 9 + 3- 5 @€&A 1 a
/ - 1 - cGA C 5 -4< /)/ + / -   +g   /.-43 
^ ^ 9 = 9 + 3- 5 Z-B<Ue J - 1  + 3V g #|1 +C(g V B< 
d 2/ / -  - 1  + 3 g[g$5@ .- g@+  < g B3B<ep + 1 / - 3-$/.- 19 B<@'/ g[ <=3/ - ./ +g  '/ g

9 + 3=- 5 9 ^ - g / + < F  = F

` A d - g < 1 -B< /.- 1=1 -$/ +9 - + / - g@+ < g -  / d  g d - VC -uB<A/ 1D+ 3A g -43 + 14#|1 +DC
g V B <XB< / @ 3=-$/.- 1D9 B<j./ g .-$/./ <Ue )€ <63 g@+ < C - 1 .- 5 \KZG\ ./A=3\GB<e / - 3-$/.- 1D9  <@./ g
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1 -A 5 / 90 \>Z- 65 B-430B<(  - g @‡ gK65  g / + < 2  + 1 / @ 9 + 3- 5Ud -- s652+ B/)/ -  g / /  /
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S - d  5B5 -4<=3 / - g  /.- 1 ZG\ g@+ <VB3- 1  <Ue / -?< + < 5 B<- 1 ./  /.- Vg - 9 + 3- 5 d 2/ + A/
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A gq|C B- d.+ B<A/Y < / @z.- g / + <  d -_e -4<- 1  5   - / - g@+ < g -  /zAV.-43q < / - 61'+ +  + 
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Vg - 9 + 3=- 5 F

h
H>MCHNM   WQ [(Q c V TIS [ QCRS  T c X Q c 5 S Q   Ge /e][Ne(f
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 = ;
q q q
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T  † p =
< 3[  /@  \GB<e 9 { =a 9 n = F ^ ^^ ^  ^C^
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1-  5 G < A 9 Z- 1 UG
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,+Dd Z - / -   +g   /.-43 g @5 1 D @g + A


5 -$/   "  < / 1'+5 .-cGA-4< g - RF + 1 # 9 =  < 9 # { =
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\ + 1 # 9 € = ZB\

: "  " 9: + ; " ; ;  " = ;  :  


9 F@{ =

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T  .B< / - 5 B<=- 1bg '-V< 1'+9 - g B<2/  5g@+ <=3V2/ + < : + :  A (g 2-4< / 5 \ 5@1 e -Y.-$/

+  ./  /.- 9( \uZ- 1 - g -43( 1D+9 : + / -4< d - d  5B5 ‡6<=3 /  / !g@+ <A/B<GA + A g@+9>+ <-4<A/ 90 \
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& ,+ 2 9 : += 2 
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