{"title":"INFORMATION THEORETICAL APPROACHES IN GAME THEORY","authors":"Seigo Kanô, Yuichi Kai","doi":"10.5109/13128","DOIUrl":"https://doi.org/10.5109/13128","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1979-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115417022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"GENERALIZED HADAMARD'S INEQUALITIES AND THEIR APPLICATIONS TO STATISTICS","authors":"A. Nishi","doi":"10.5109/13129","DOIUrl":"https://doi.org/10.5109/13129","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1979-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134598718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A TRUNCATED PLAY-THE-WINNER PROCEDURE FOR SELECTING THE BEST OF $ k geqq 3 $ BINOMIAL POPULATION","authors":"K. Schriever","doi":"10.5109/13121","DOIUrl":"https://doi.org/10.5109/13121","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131845145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let an : n 1} be a sequence of i.i.d. Banach space valued random variables with E[X„]=0 and Ell X.112<00, and let So=0, Sn= XiF X2+ . . . + Xn, n 1. We prove that if {Sn : n_. 1} satisfies the LIL in B then the sequence {77,, : n .1} satisfies the LIL in C([0, 1], B), where 77n(t)=S[nt]+ (nt—[nt]) X[nt]-14, Ot51 and C([°, 1], B) --={ f : [0, 1] ----. BI f is continuous}. We also use this result to give an alternative to the proof of the LIL of Brownian motion in Banach spaces.
设an: n 1}是一个序列，其中E[X "]=0且Ell X.112<00的Banach空间值随机变量，设So=0, Sn= XiF X2+…+ Xn n 1。我们证明如果{Sn: n_。1}满足B中的LIL，则序列{77，，:n .1}满足C([0,1]， B)中的LIL，其中77n(t)=S[nt]+ (nt - [nt]) X[nt]-14, Ot51和C([°，1]，B)—={f:[0,1] ----。BI f是连续的。我们还利用这一结果给出了巴拿赫空间中布朗运动LIL的另一种证明。
{"title":"A RANDOM WALK AND ITS LIL IN A BANACH SPACE","authors":"M. Chang","doi":"10.5109/13125","DOIUrl":"https://doi.org/10.5109/13125","url":null,"abstract":"Let an : n 1} be a sequence of i.i.d. Banach space valued random variables with E[X„]=0 and Ell X.112<00, and let So=0, Sn= XiF X2+ . . . + Xn, n 1. We prove that if {Sn : n_. 1} satisfies the LIL in B then the sequence {77,, : n .1} satisfies the LIL in C([0, 1], B), where 77n(t)=S[nt]+ (nt—[nt]) X[nt]-14, Ot51 and C([°, 1], B) --={ f : [0, 1] ----. BI f is continuous}. We also use this result to give an alternative to the proof of the LIL of Brownian motion in Banach spaces.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117261550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We shall be concerned with the optimization problem of semi-Markov decision processes with countable state space and compact action space. Defined is the generalized reward function associated with the semi-Markov decision processes which include the ordinary discounted Markov decision processes of discrete time parameter and also the continuous time Markov decision processes. Main results are (a) the existence of an optimal stationary policy and (b) the relation between the maximal expected reward and the optimality equation. Also (c) some properties of the optimal staionary policy and the principle of optimality are obtained.
{"title":"SEMI-MARKOV DECISION PROCESSES WITH COUNTABLE STATE SPACE AND COMPACT ACTION SPACE","authors":"M. Yasuda","doi":"10.5109/13122","DOIUrl":"https://doi.org/10.5109/13122","url":null,"abstract":"We shall be concerned with the optimization problem of semi-Markov decision processes with countable state space and compact action space. Defined is the generalized reward function associated with the semi-Markov decision processes which include the ordinary discounted Markov decision processes of discrete time parameter and also the continuous time Markov decision processes. Main results are (a) the existence of an optimal stationary policy and (b) the relation between the maximal expected reward and the optimality equation. Also (c) some properties of the optimal staionary policy and the principle of optimality are obtained.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"60 36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133131772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"RATES OF CONVERGENCE IN CENTRAL LIMIT THEOREM FOR MARTINGALE DIFFERENCES","authors":"Yutaka Kato","doi":"10.5109/13119","DOIUrl":"https://doi.org/10.5109/13119","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130594998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"STOCHASTIC GAME MODELS FOR THE DETERMINATION OF THE OPTIMAL CONTINUOUS SAMPLING INSPECTION PLANS","authors":"T. Sakamoto, M. Kurano","doi":"10.5109/13120","DOIUrl":"https://doi.org/10.5109/13120","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130924093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper the limit probability and the total deviation are considered by introducing an artificial transition matrix in Markov jump processes. Section 2 contains a simultaneous equation which the limit probability satisfies. In a single positive recurrent class the simultaneous equation can be reduced to an ordinary one and its solution has been given by Ballow [1], Miller [11] and Feller [5]. We note that the calculation has relation to summability methods. If the state is finite, then we can get an explicit formula of the limit probability for Markov jump processes with several classes by solving the simultaneous equation. In section 3 we shall define a total deviation from the limit probability. Our results extend that of Kemeny and Snell [9] to the denumerable state case. The notion, deviation measure, in [9] is utilized for Markov decision processes (Veinott [13]).
{"title":"THE CALUCULATION OF LIMIT PROBABILITIES FOR MARKOV JUMP PROCESSES","authors":"M. Yasuda","doi":"10.5109/13124","DOIUrl":"https://doi.org/10.5109/13124","url":null,"abstract":"In this paper the limit probability and the total deviation are considered by introducing an artificial transition matrix in Markov jump processes. Section 2 contains a simultaneous equation which the limit probability satisfies. In a single positive recurrent class the simultaneous equation can be reduced to an ordinary one and its solution has been given by Ballow [1], Miller [11] and Feller [5]. We note that the calculation has relation to summability methods. If the state is finite, then we can get an explicit formula of the limit probability for Markov jump processes with several classes by solving the simultaneous equation. In section 3 we shall define a total deviation from the limit probability. Our results extend that of Kemeny and Snell [9] to the denumerable state case. The notion, deviation measure, in [9] is utilized for Markov decision processes (Veinott [13]).","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125464181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A connection between Markov Decision Process (MDP) and Markov potential theory has two sides. One is the potential theoritic development of MDP and the other is the alternative proof of the results in MDP owing to Markov potential theory. Shaufele [12] belongs to the later, but it seems interesting from the standpoint of the mathematical programming to establish the development of MDP by using certain potential notion. Several approaches have been tried. Watanabe [16] interpreted the monotonicity of Howard's iteration [8] in the relation to the a dual problem of Linear Programming. By the property of a potential kernel, Furukawa [6] and Aso and Kimura [1] proved a policy improvement. A formulation of MDP by potential theoretic notion has been tried by Hordijk [7]. In many cases it is restricted to a transient potential theory because its analysis is simpler. In this paper we shall define a new potential in order to serve a general policy improvement. Our aim is to expose theorems which are available to several cases of MDP. By the potential theoretic terms, we can interpret policy improvements of MDP as follows ; The increase of rewards in MDP consists of the potential with a charge of an increment of the policy improvement and a regular function. If it is transient, then the potential is reduced to the ordinary one and the regular function equals zero. Hence this consists with that of Watanabe [16]. The merit of the potential is that it connects the policy improvement with the increment of rewards. We shall consider the following cost criteria of MDP ; (1) discounted case, (2) average case, (3) nearly optimal case and (4) sensitive discounted case. Case (1) and (2) are representitive and discussed by many authors. Especially we list up Howard [7] and Blackwell [2], [3] for (1) and Howard [8] and Derman [4], [5] for (2). Case (3) is due to Blackwell [2]. Extending case (3), case (4) is studied by Miller and Veinott [11] and Veinott [14], [15].
{"title":"POLICY IMPROVEMENT IN MARKOV DECISION PROCESSES AND MARKOV POTENTIAL THEORY","authors":"M. Yasuda","doi":"10.5109/13123","DOIUrl":"https://doi.org/10.5109/13123","url":null,"abstract":"A connection between Markov Decision Process (MDP) and Markov potential theory has two sides. One is the potential theoritic development of MDP and the other is the alternative proof of the results in MDP owing to Markov potential theory. Shaufele [12] belongs to the later, but it seems interesting from the standpoint of the mathematical programming to establish the development of MDP by using certain potential notion. Several approaches have been tried. Watanabe [16] interpreted the monotonicity of Howard's iteration [8] in the relation to the a dual problem of Linear Programming. By the property of a potential kernel, Furukawa [6] and Aso and Kimura [1] proved a policy improvement. A formulation of MDP by potential theoretic notion has been tried by Hordijk [7]. In many cases it is restricted to a transient potential theory because its analysis is simpler. In this paper we shall define a new potential in order to serve a general policy improvement. Our aim is to expose theorems which are available to several cases of MDP. By the potential theoretic terms, we can interpret policy improvements of MDP as follows ; The increase of rewards in MDP consists of the potential with a charge of an increment of the policy improvement and a regular function. If it is transient, then the potential is reduced to the ordinary one and the regular function equals zero. Hence this consists with that of Watanabe [16]. The merit of the potential is that it connects the policy improvement with the increment of rewards. We shall consider the following cost criteria of MDP ; (1) discounted case, (2) average case, (3) nearly optimal case and (4) sensitive discounted case. Case (1) and (2) are representitive and discussed by many authors. Especially we list up Howard [7] and Blackwell [2], [3] for (1) and Howard [8] and Derman [4], [5] for (2). Case (3) is due to Blackwell [2]. Extending case (3), case (4) is studied by Miller and Veinott [11] and Veinott [14], [15].","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"164 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1978-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129355215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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