On regularity conditions for random fields
Author:
Richard C. Bradley
Journal:
Proc. Amer. Math. Soc. 121 (1994), 593598
MSC:
Primary 60G60; Secondary 28D15, 60G25
MathSciNet review:
1219721
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Indexed by the integer lattice of dimension at least two, there exists a nondegenerate strictly stationary random field which is onedependent with respect to "latticehalfspaces" but which is also measurable with respect to its own tail sigmafield.
 [1]
Richard
C. Bradley, A bilaterally deterministic
𝜌mixing stationary random sequence, Trans. Amer. Math. Soc. 294 (1986), no. 1, 233–241. MR 819945
(87h:60080), http://dx.doi.org/10.1090/S00029947198608199450
 [2]
Richard
C. Bradley, A caution on mixing conditions for random fields,
Statist. Probab. Lett. 8 (1989), no. 5,
489–491. MR 1040812
(91h:60052), http://dx.doi.org/10.1016/01677152(89)900321
 [3]
Richard
C. Bradley, On the spectral density and asymptotic normality of
weakly dependent random fields, J. Theoret. Probab. 5
(1992), no. 2, 355–373. MR 1157990
(93e:60094), http://dx.doi.org/10.1007/BF01046741
 [4]
, Equivalent mixing conditions for random fields, Ann. Probab. (in press).
 [5]
A.
V. Bulinskiĭ, Various mixing conditions and the asymptotic
normality of random fields, Dokl. Akad. Nauk SSSR 299
(1988), no. 4, 785–789 (Russian); English transl., Soviet Math.
Dokl. 37 (1988), no. 2, 443–448. MR 943731
(89i:60104)
 [6]
, Limit theorems under weak dependence conditions, Moscow Univ. Press, Moscow, 1989. (Russian)
 [7]
Robert
M. Burton, Manfred
Denker, and Meir
Smorodinsky, Finite state bilaterally deterministic strongly mixing
processes, Israel J. Math. 95 (1996), 115–133.
MR
1418290 (98b:60069), http://dx.doi.org/10.1007/BF02761036
 [8]
R.
L. Dobrušin, Description of a random field by means of
conditional probabilities and conditions for its regularity, Teor.
Verojatnost. i Primenen 13 (1968), 201–229 (Russian,
with English summary). MR 0231434
(37 #6989)
 [9]
V. F. Gaposhkin, Moment bounds for integrals of mixing fields, Theory Probab. Appl. 36 (1991), 249260.
 [10]
Charles
M. Goldie and Priscilla
E. Greenwood, Variance of setindexed sums of mixing random
variables and weak convergence of setindexed processes, Ann. Probab.
14 (1986), no. 3, 817–839. MR 841586
(88e:60038b)
 [11]
V. V. Gorodetskii, The central limit theorem and an invariance principle for weakly dependent random fields, Soviet Math. Dokl. 29 (1984), 529532.
 [12]
V. M. Gurevic, On onesided and twosided regularity of stationary random processes, Soviet Math. Dokl. 14 (1973), 804808.
 [13]
Carla
C. Neaderhouser, An almost sure invariance principle for partial
sums associated with a random field, Stochastic Process. Appl.
11 (1981), no. 1, 1–10. MR 608003
(82h:60059), http://dx.doi.org/10.1016/03044149(81)90017X
 [14]
Richard
A. Olshen, The coincidence of measure algebras under an
exchangeable probability., Z. Wahrscheinlichkeitstheorie und Verw.
Gebiete 18 (1971), 153–158. MR 0288797
(44 #5992)
 [15]
Donald
S. Ornstein and Benjamin
Weiss, Every transformation is bilaterally deterministic,
Israel J. Math. 21 (1975), no. 23, 154–158.
Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974).
MR
0382600 (52 #3482)
 [16]
Murray
Rosenblatt, Stationary sequences and random fields,
Birkhäuser Boston, Inc., Boston, MA, 1985. MR 885090
(88c:60077)
 [17]
Lanh
Tat Tran, Kernel density estimation on random fields, J.
Multivariate Anal. 34 (1990), no. 1, 37–53. MR 1062546
(91j:62050), http://dx.doi.org/10.1016/0047259X(90)90059Q
 [18]
I. G. Zhurbenko, On mixing conditions for random processes with values in a Hilbert space, Soviet Math. Dokl. 30 (1984), 465467.
 [19]
I.
G. Žurbenko, The spectral analysis of time series,
NorthHolland Series in Statistics and Probability, vol. 2,
NorthHolland Publishing Co., Amsterdam, 1986. With a foreword by A. N.
Kolmogorov. MR
860209 (87m:62273)
 [1]
 R. C. Bradley, A bilaterally deterministic mixing stationary random sequence, Trans. Amer. Math. Soc. 294 (1986), 233241. MR 819945 (87h:60080)
 [2]
 , A caution on mixing conditions for random fields, Statist. Probab. Lett. 8 (1989), 489491. MR 1040812 (91h:60052)
 [3]
 , On the spectral density and asymptotic normality of weakly dependent random fields, J. Theoret. Probab. 5 (1992), 355373. MR 1157990 (93e:60094)
 [4]
 , Equivalent mixing conditions for random fields, Ann. Probab. (in press).
 [5]
 A. V. Bulinskii, On various conditions of mixing and asymptotic normality of random fields, Soviet Math. Dokl. 37 (1988), 443448. MR 943731 (89i:60104)
 [6]
 , Limit theorems under weak dependence conditions, Moscow Univ. Press, Moscow, 1989. (Russian)
 [7]
 R. Burton, M. Denker, and M. Smorodinsky, Finite state bilaterally deterministic strongly mixing processes, Israel J. Math. (submitted). MR 1418290 (98b:60069)
 [8]
 R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity, Theory Probab. Appl. 13 (1968), 197224. MR 0231434 (37:6989)
 [9]
 V. F. Gaposhkin, Moment bounds for integrals of mixing fields, Theory Probab. Appl. 36 (1991), 249260.
 [10]
 C. M. Goldie and P. Greenwood, Variance of setindexed sums of mixing random variables and weak convergence of setindexed processes, Ann. Probab. 14 (1986), 817839. MR 841586 (88e:60038b)
 [11]
 V. V. Gorodetskii, The central limit theorem and an invariance principle for weakly dependent random fields, Soviet Math. Dokl. 29 (1984), 529532.
 [12]
 V. M. Gurevic, On onesided and twosided regularity of stationary random processes, Soviet Math. Dokl. 14 (1973), 804808.
 [13]
 C. C. Neaderhouser, An almost sure invariance principle for partial sums associated with a random field, Stochastic Process. Appl. 11 (1981), 110. MR 608003 (82h:60059)
 [14]
 R. A. Olshen, The coincidence of measure algebras under an exchangable probability, Z. Wahrsch. Verw. Gebiete 18 (1971), 153158. MR 0288797 (44:5992)
 [15]
 D. S. Ornstein and B. Weiss, Every transformation is bilaterally deterministic, Israel J. Math. 21 (1975), 154158. MR 0382600 (52:3482)
 [16]
 M. Rosenblatt, Stationary sequences and random fields, Birkhäuser, Boston, MA, 1985. MR 885090 (88c:60077)
 [17]
 L. T. Tran, Kernel density estimation on random fields, J. Multivariate Anal. 34 (1990), 3753. MR 1062546 (91j:62050)
 [18]
 I. G. Zhurbenko, On mixing conditions for random processes with values in a Hilbert space, Soviet Math. Dokl. 30 (1984), 465467.
 [19]
 , The spectral analysis of time series, NorthHolland, Amsterdam, 1986. MR 860209 (87m:62273)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412197213
PII:
S 00029939(1994)12197213
Keywords:
Onedependent,
deterministic,
strictly stationary random field
Article copyright:
© Copyright 1994
American Mathematical Society
