[1] K. L. Chung :
“Contributions to the theory of Markov chains ,”
J. Research Nat. Bur. Standards
50 : 4
(April 1953 ),
pp. 203–208 .
Research Paper 2411.
MR
0055608
Zbl
0053.27201
article

Abstract
BibTeX

The fundamentals of the theory of denumerable Markov chains with stationary transition probabilities were laid down by Kolmogorov and further work was done by Doblin. The theory of recurrent events of Feller is closely related, if not coextensive. Some new results obtained by T. E. Harris turn out to tie up very nicely with some amplifications of Doblin’s work. Harris was led to consider the probabilities of hitting one state before another, starting from a third one. This idea of considering three states, one initial, one “taboo”, and one final, is more fully developed in the present work. The notion of first passage time of the “union” or “intersection” of two states is also introduced here. The interplay between these notions is illustrated.

@article {key0055608m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Contributions to the theory of {M}arkov
chains},
JOURNAL = {J. Research Nat. Bur. Standards},
FJOURNAL = {Journal of Research of the National
Bureau of Standards},
VOLUME = {50},
NUMBER = {4},
MONTH = {April},
YEAR = {1953},
PAGES = {203--208},
NOTE = {Research Paper 2411. MR:0055608. Zbl:0053.27201.},
ISSN = {0160-1741},
}
[2] K. L. Chung :
“Contributions to the theory of Markov chains, II ,”
Trans. Amer. Math. Soc.
76
(1954 ),
pp. 397–419 .
MR
0063603
Zbl
0058.34602
article

BibTeX
@article {key0063603m,
AUTHOR = {Chung, K. L.},
TITLE = {Contributions to the theory of {M}arkov
chains, {II}},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {76},
YEAR = {1954},
PAGES = {397--419},
DOI = {10.2307/1990789},
NOTE = {MR:0063603. Zbl:0058.34602.},
ISSN = {0002-9947},
}
[3] K. L. Chung :
“Foundations of the theory of continuous parameter Markov chains ,”
pp. 29–40
in
Proceedings of the third Berkeley symposium on mathematical statistics and probability, 1954–1955 ,
vol. II: Contributions to probability theory .
Edited by J. Neyman .
University of California Press (Berkeley, CA and Los Angeles, CA ),
1956 .
MR
0084884
Zbl
0071.35001
inproceedings

Abstract
People
BibTeX

This paper is exclusively concerned with continuous parameter Markov processes with a denumerably infinite number of states and stationary transition matrix function. The foundations of the proper theory of such processes, as distinguished from that of the discrete parameter version, or of Markov processes which are either more special (for example, general state space; nonstationary transition matrix function), were laid by Doob [1942; 1945], and Lévy [1951; 1952; 1953]. Roughly speaking it was Lévy in 1952 who drew, in his inimitable way, the comprehensive picture while Doob, ten years earlier, had supplied the essential ingredients. The present effor aims at a synthesis of the most fundamental parts of the theory, made possible by the contributions of these two authors. While the results given here generally extend and clarify those in the cited literature, immense credit must go to Professors Lévy and Doob for the inspiration of their pioneer work. To them I am also indebted for much valuable discussion through correspondence and conversation. An attempt is made in the presentation to be quite formal and rigorous, in the spirit of Doob’s already classic treatise [1953]. Further developments of the theory will be published elsewhere.

@inproceedings {key0084884m,
AUTHOR = {Chung, K. L.},
TITLE = {Foundations of the theory of continuous
parameter {M}arkov chains},
BOOKTITLE = {Proceedings of the third {B}erkeley
symposium on mathematical statistics
and probability, 1954--1955},
EDITOR = {J. Neyman},
VOLUME = {II: Contributions to probability theory},
PUBLISHER = {University of California Press},
ADDRESS = {Berkeley, CA and Los Angeles, CA},
YEAR = {1956},
PAGES = {29--40},
URL = {http://projecteuclid.org/euclid.bsmsp/1200502004},
NOTE = {MR:0084884. Zbl:0071.35001.},
}
[4] K. L. Chung :
“Some new developments in Markov chains ,”
Trans. Amer. Math. Soc.
81
(1956 ),
pp. 195–210 .
MR
0075481
Zbl
0075.14001
article

Abstract
BibTeX

The purpose of this paper is to point out some new connections between the sample function behavior and the analytical properties of the transition probability functions of a continuous parameter Markov chain with stationary transition probabilities. The main idea is that of a post-exit process derived from the original process by considering its evolution after the exit from a stable state. This leads to various relations between conditional probabilities all of which are “intuitively obvious” but require sometimes painstaking proofs if probability theory like any other branch of mathematics is to be treated as a discipline in logic. These relations imply certain analytical properties of the transition functions, obtained recently by D. G. Austin by purely analytical means, and exhibit them in connection with other quantities introduced in this paper. They also complete some results due to Doob and Lévy. The well-known differential equations of Kolmogorov are seen to be limiting cases of certain more generally valid differential equations involving a continuous parameter. One of these expresses the fundamental transition property of the post-exit process, and the other a similar property of the renewal density of an imbedded renewal process.

@article {key0075481m,
AUTHOR = {Chung, K. L.},
TITLE = {Some new developments in {M}arkov chains},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {81},
YEAR = {1956},
PAGES = {195--210},
DOI = {10.2307/1992859},
NOTE = {MR:0075481. Zbl:0075.14001.},
ISSN = {0002-9947},
}
[5] K. L. Chung :
“Some aspects of continuous parameter Markov chains ,”
Publ. Inst. Statist. Univ. Paris
6
(1957 ),
pp. 271–287 .
MR
0102128
Zbl
0082.34502
article

BibTeX
@article {key0102128m,
AUTHOR = {Chung, K. L.},
TITLE = {Some aspects of continuous parameter
{M}arkov chains},
JOURNAL = {Publ. Inst. Statist. Univ. Paris},
FJOURNAL = {Publications de l'Institut de Statistique
de l'Universit\'e de Paris},
VOLUME = {6},
YEAR = {1957},
PAGES = {271--287},
NOTE = {MR:0102128. Zbl:0082.34502.},
ISSN = {0553-2930},
}
[6] K. L. Chung :
“On a basic property of Markov chains ,”
Ann. of Math. (2)
68 : 1
(July 1958 ),
pp. 126–149 .
MR
0100924
Zbl
0228.60023
article

Abstract
BibTeX

The main question discussed in this paper has been in the air for some time. Roughly speaking it may be stated as follows: Given a continuous parameter Markov process \( \{x_t,\ t\geq 0\} \) with stationary transition probabilities, let us start it off at a random time \( \alpha \) chosen according to the way it has been unravelling, as it were, but without prevision of the future. Does the shifted process \( \{x_{\alpha+t},\ t\geq 0\} \) remain Markovian with the same transition probabilities, and, knowing \( x_{\alpha} \) , is the past \( \{x_t,\ t\leq\alpha\} \) independent of the future \( \{x_t,\ t\geq\alpha\} \) ?

@article {key0100924m,
AUTHOR = {Chung, K. L.},
TITLE = {On a basic property of {M}arkov chains},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {68},
NUMBER = {1},
MONTH = {July},
YEAR = {1958},
PAGES = {126--149},
DOI = {10.2307/1970046},
NOTE = {MR:0100924. Zbl:0228.60023.},
ISSN = {0003-486X},
}
[7] K. L. Chung :
“Some remarks on taboo probabilities ,”
Illinois J. Math.
5 : 3
(1961 ),
pp. 431–435 .
MR
0131902
Zbl
0101.11601
article

BibTeX
@article {key0131902m,
AUTHOR = {Chung, K. L.},
TITLE = {Some remarks on taboo probabilities},
JOURNAL = {Illinois J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {5},
NUMBER = {3},
YEAR = {1961},
PAGES = {431--435},
URL = {http://projecteuclid.org/euclid.ijm/1255630889},
NOTE = {MR:0131902. Zbl:0101.11601.},
ISSN = {0019-2082},
}
[8] K. L. Chung :
“Probabilistic methods in Markov chains ,”
pp. 35–56
in
Proceedings of the fourth Berkeley symposium on mathematical statistics and probability ,
vol. II .
Edited by J. Neyman .
University of California Press (Berkeley, CA ),
1961 .
MR
0131901
Zbl
0106.33303
incollection

People
BibTeX
@incollection {key0131901m,
AUTHOR = {Chung, K. L.},
TITLE = {Probabilistic methods in {M}arkov chains},
BOOKTITLE = {Proceedings of the fourth {B}erkeley
symposium on mathematical statistics
and probability},
EDITOR = {J. Neyman},
VOLUME = {II},
PUBLISHER = {University of California Press},
ADDRESS = {Berkeley, CA},
YEAR = {1961},
PAGES = {35--56},
URL = {http://projecteuclid.org/euclid.bsmsp/1200512592},
NOTE = {MR:0131901. Zbl:0106.33303.},
}
[9] K. L. Chung :
“On the Martin boundary for Markov chains ,”
Proc. Nat. Acad. Sci. U.S.A.
48 : 6
(June 1962 ),
pp. 963–968 .
MR
0145584
Zbl
0228.60030
article

Abstract
BibTeX

The Martin boundary for a discrete parameter Markov chain was first considered by Doob, using the theory of R. S. Martin after whom the boundary is named. A direct and ingenious method was later found by Hunt, who also strengthened Doob’s results in several points. In this note, I shall sketch a natural approach to the theory in the continuous parameter case.

@article {key0145584m,
AUTHOR = {Chung, K. L.},
TITLE = {On the {M}artin boundary for {M}arkov
chains},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {48},
NUMBER = {6},
MONTH = {June},
YEAR = {1962},
PAGES = {963--968},
URL = {http://www.jstor.org/stable/71264},
NOTE = {MR:0145584. Zbl:0228.60030.},
ISSN = {0027-8424},
}
[10] K. L. Chung :
“On the boundary theory for Markov chains ,”
Acta Math.
110
(1963 ),
pp. 19–77 .
MR
0150829
Zbl
0292.60121
article

BibTeX
@article {key0150829m,
AUTHOR = {Chung, Kai Lai},
TITLE = {On the boundary theory for {M}arkov
chains},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {110},
YEAR = {1963},
PAGES = {19--77},
DOI = {10.1007/BF02391854},
NOTE = {MR:0150829. Zbl:0292.60121.},
ISSN = {0001-5962},
}
[11] K. L. Chung :
“On the boundary theory for Markov chains, II ,”
Acta Math.
115 : 1
(1966 ),
pp. 111–163 .
MR
0189117
Zbl
0315.60041
article

Abstract
BibTeX

This paper may be regarded as a new and fairly self-contained one attached to §§2–4 of [Chung 1963]. These sections are entitled “Terminology and notation”, “The boundary” and “Fundamental theorems” respectively. The rest of [Chung 1963] is either contained in a more general treatment (§5, §9 and parts of §6), or may be set aside as special cases under additional hypotheses (parts of §6, §7 and §8). In particular the whole idea of “dual boundary” is dispensed with here, though this is not to say it should be abandoned forever. Reference to [Chung 1963] beyond §4 will be pinpointed.

@article {key0189117m,
AUTHOR = {Chung, Kai Lai},
TITLE = {On the boundary theory for {M}arkov
chains, {II}},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {115},
NUMBER = {1},
YEAR = {1966},
PAGES = {111--163},
DOI = {10.1007/BF02392205},
NOTE = {MR:0189117. Zbl:0315.60041.},
ISSN = {0001-5962},
}
[12] K. L. Chung :
“Boundary behavior of Markov chains and its contributions to general processes ,”
pp. 499–505
in
Actes du Congrès International des Mathématiciens
(Nice, France, September 1–10, 1970 ),
Tome 2 .
Gauthier-Villars (Paris ),
1971 .
MR
0420882
Zbl
0298.60047
incollection

BibTeX
@incollection {key0420882m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Boundary behavior of {M}arkov chains
and its contributions to general processes},
BOOKTITLE = {Actes du {C}ongr\`es {I}nternational
des {M}ath\'ematiciens},
VOLUME = {2},
PUBLISHER = {Gauthier-Villars},
ADDRESS = {Paris},
YEAR = {1971},
PAGES = {499--505},
NOTE = {(Nice, France, September 1--10, 1970).
MR:0420882. Zbl:0298.60047.},
}
[13] K. L. Chung :
“‘Markov chain must have a beginning.’ In memory of Prof. Loo Keng Hua ,”
J. Math. Res. Expo.
1
(1986 ),
pp. 1–3 .
MR
858088
Zbl
0632.60064
article

People
BibTeX
@article {key858088m,
AUTHOR = {Chung, K. L.},
TITLE = {``{M}arkov chain must have a beginning.''
{I}n memory of {P}rof.\ {L}oo {K}eng
{H}ua},
JOURNAL = {J. Math. Res. Expo.},
FJOURNAL = {Journal of Mathematical Research and
Exposition (English Edition)},
NUMBER = {1},
YEAR = {1986},
PAGES = {1--3},
NOTE = {MR:858088. Zbl:0632.60064.},
ISSN = {ISSN 1000-341X},
}