D:\sbornik\...\Naira.DVI Mathematical Problems of Computer Science 29, 2007, 89{96. On Reliability Appr oach for T esting of M any Distr ibutions for P air of M ar kov Chains E vg u e n i H a r o u t u n ia n a n d N a ir a Gr ig o r ya n Institue for Informatics and Automation Problems of NAS of RA e-mail: evhar@ipia.sci.am Abstract The problem of three hypotheses logarithmical asymptotically optimal testing for a pair of simple homogeneous stationary Markov chains is examined. It is supposed that M probability distributions are known and each of Markov chains independently of other follows to one of them. The matrix of all error probability exponents (reliabilities) is studied. Refer ences [1 ] R . F. A h ls we d e a n d E . A . H a r o u t u n ia n , \ On lo g a r it h m ic a lly a s ym p t o t ic a lly o p t i- m a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " , L ecture Notes in Computer Science, vo l. 4 1 2 3 ,\ Ge n e r a lTh e o r y o f In fo r m a t io n Tr a n s fe r a n d Co m b in a t o r ic s " , S p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [2 ] E . A . H a r o u t u n ia n , \ L o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a - t is t ic a l h yp o t h e s e s " , P roblems of Control and Information Theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 -4 2 1 , 1 9 9 0 . [3 ] S . N a t a r a ja n , \ L a r g e d e via t io n s , h yp o t h e s e s t e s t in g , a n d s o u r c e c o d in g fo r ¯ n it e Ma r ko v c h a in s " ,IE E E Trans. Inform. Theory, vo l 3 1 , n o . 3 , p p . 3 6 0 -3 6 5 , 1 9 8 5 . [4 ] E . A . H a r o u t u n ia n , \ On a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s c o n c e r n in g Ma r ko v c h a in " , Izvestiya Akademii Nauk Armenii, M athematika,( in R u s s ia n ) , vo l. 2 3 , n o . 1 , p p . 7 6 -8 0 , 1 9 8 8 . [5 ] E . A . H a r o u t u n ia n , \ On a s ym p t o t ic a lly o p t im a l c r it e r ia fo r Ma r ko v c h a in s " , ( in R u s - s ia n ) , F irst W orld Congress of B ernoulli Society, s e c t io n 2 , vo l. 2 , n o . 3 , p p . 1 5 3 -1 5 6 , 1 9 8 9 . [6 ] M. Gu t m a n , \ A s ym p t o t ic a lly o p t im a l c la s s ī c a t io n fo r m u lt ip le t e s t s wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s " , IE E E Tr a n s . In fo r m . Th e o r y, vo l 3 5 , n o 2 , Ma r c h , 4 0 1 -4 0 8 , 1 9 8 9 . [7 ] E . A . H a r o u t u n ia n , \ A s ym p t o t ic a lly o p t im a l t e s t in g o f m a n y s t a t is t ic a l h yp o t h e s e s c o n c e r n in g Ma r ko v c h a in " , ( in R u s s ia n ) , 5-th Intern. Vilnius Conferance on P robability Theory and M athem. Statistics, vo l. 1 ( A -L ) , p p . 2 0 2 -2 0 3 , 1 9 8 9 . 8 9 9 0 On Reliability Approach for Testing of Many Distributions for Pair of Markov Chains [8 ] E . A . H a r o u t u n ia n , M. E . H a r o u t u n ia n a n d A . N . H a r u t yu n ya n , \ R e lia b ilit y c r it e - r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , 1 6 0 p , A c c e p t e d fo r p u b lic a t io n in Fo u n d a t io n a n d Tr e n d s in Co m u n ic a t io n s a n d In fo r m a t io n Th e o r y. [9 ] E . A . H a r o u t u n ia n a n d P . M. H a ko b ya n , \ On lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s e s t e s t in g o f t h r e e d is t r ib u t io n s fo r p a ir o f o b je c t s " , M athematical P roblems of Computer Science, vo l. 2 4 , p p . 7 6 { 8 1 , 2 0 0 5 . [1 0 ] A . U lu b a b ya n \ On lo g a r it h m ic a lly a s yp t o t ic a lly o p t im a l h yp o t h e s e s t e s t in g o f t wo d is t r ib u t io n s fo r p a ir o f in d e p e n d e n t o b je c t s o f m a r ko v c h a in " , Thesis of bachelor, State E nginering University of Armenia 2 0 0 5 . [1 1 ] I. Cs is z ¶a r a n d J. K ¶o r n e r , \ In fo r m a t io n t h e o r y, c o d in g t h e o r e m s fo r d is c r e t e m e m o r yle s s s ys t e m s " , Academic P ress, New York, 1 9 8 1 . ºñÏáõ Ù³ñÏáíÛ³Ý ßÕóݻñÇ Ýϳïٳٵ µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ ëïáõ·Ù³Ý Ñáõë³ÉÇáõÃÛ³Ý ëϽµáõÝùÇ Ù³ëÇÝ º. ². гñáõÃÛáõÝÛ³Ý ¨ Ü. ². ¶ñÇ·áñÛ³Ý ²Ù÷á÷áõÙ àõëáõÙݳëÇñí»É ¿ »ñÏáõ ³ÝÏ³Ë ëï³óÇáݳñ Ù³ñÏáíÛ³Ý ßÕóݻñÇó ϳ½Ùí³Í Ùá¹»ÉÇ Ñ³Ù³ñ µ³½Ù³ÏÇ í³ñϳÍÝ»ñÇ ëïáõ·Ù³Ý ËݹÇñÁ: M ( ¸ 2 ) ѳí³Ý³Ï³ÝáõÃÛáõÝÝ»ñÇ µ³ßËáõÙÝ»ñÁ ѳÛïÝÇ »Ý: ²ÝÏ³Ë ûµÛ»ÏïÝ»ñÇó Ûáõñ³ù³ÝãÛáõñÁ µ³ßËí³Í ¿ Áëï ¹ñ³ÝóÇó Ù»ÏÇ: àõëáõÙݳëÇñí³Í ¿ ë˳ÉÝ»ñÇ Ñ³í³Ý³Ï³ÝáõÃÛáõÝÝ»ñÇ óáõóÇãÝ»ñÇ (Ñáõë³ÉÇáõÃÛáõÝÝ»ñÇ) ½áõÛ·»ñÇ Ù³ïñÇóÁ: