February 2019, 39(2): 667-682. doi: 10.3934/dcds.2019027

Stochastic dominance for shift-invariant measures

Queen Mary University of London, Mile End Road, London, E1 4NS, UK

Received  April 2016 Revised  July 2018 Published  November 2018

Fund Project: The author was partially supported by EPSRC grant EP/L02246X/1.

Let $X$ be the full shift on two symbols. The lexicographic order induces a partial order known as first-order stochastic dominance on the collection ${\mathcal{M}}_{X}$ of its shift-invariant probability measures. We present a study of the fine structure of this dominance order, denoted by $\prec$, and give criteria for establishing comparability or incomparability between measures in ${\mathcal{M}}_{X}$. The criteria also give an insight to the complicated combinatorics of orbits in the shift. As a by-product, we give a direct proof that Sturmian measures are totally ordered with respect to $\prec$.

Citation: Vasso Anagnostopoulou. Stochastic dominance for shift-invariant measures. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 667-682. doi: 10.3934/dcds.2019027
References:
[1]

J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.

[2]

V. Anagnostopoulou, Sturmian Measures and Stochastic Dominance in Ergodic Optimization, Ph.D. thesis, Queen Mary University of London, 2009.

[3]

V. Anagnostopoulou and O. Jenkinson, Which beta-shifts have a largest invariant measure?, Jour. Lon. Math. Soc., 79 (2009), 445-464. doi: 10.1112/jlms/jdn070.

[4]

J. Berstel and P. Séébold, Sturmian words, in Algebraic Combinatorics on Words (Encyclopaedia of Mathematics and its Applications 90), (M. Lothaire), Cambridge University Press, (2002), 45–110.

[5]

T. Bousch, Le poisson n'a pas d'arȇtes, Ann. Inst. Henri Poincaré (Proba. et Stat.), 36 (2000), 489-508. doi: 10.1016/S0246-0203(00)00132-1.

[6]

T. Bousch, Une propriété de domination convexe pour les orbites sturmiennes, Can. Jour. Math., 67 (2015), 90-106. doi: 10.4153/CJM-2014-009-8.

[7]

T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, tetris heaps, and the finiteness conjecture, Jour. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2.

[8]

S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Camb. Phil. Soc., 115 (1994), 451-481. doi: 10.1017/S0305004100072236.

[9]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford University Press, 1979.

[10]

O. Jenkinson, Conjugacy Rigidity, Cohomological Triviality, and Barycentres of Invariant Measures, Ph.D. thesis, University of Warwick, 1996.

[11]

O. Jenkinson, Frequency locking on the boundary of the barycentre set, Exp. Math., 9 (2000), 309-317.

[12]

O. Jenkinson, Maximum hitting frequency and fastest mean return time, Nonlinearity, 18 (2005), 2305-2321. doi: 10.1088/0951-7715/18/5/022.

[13]

O. Jenkinson, Ergodic optimization, Discrete & Cont. Dyn. Sys., 15 (2006), 197-224. doi: 10.3934/dcds.2006.15.197.

[14]

O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1-12. doi: 10.1090/S1079-6762-07-00170-9.

[15]

O. Jenkinson, A partial order on $× 2$-invariant measures, Math. Res. Lett., 15 (2008), 893-900. doi: 10.4310/MRL.2008.v15.n5.a6.

[16]

T. KamaeU. Krengel and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Prob., 5 (1977), 899-912.

[17]

H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, 3rd edition, Springer, 2016. doi: 10.1007/978-3-319-21708-6.

[18]

T. Lindvall, On Strassen's theorem on stochastic domination, Electron. Commun. Probab., 4 (1999), 51-59. doi: 10.1214/ECP.v4-1005.

[19]

M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[20]

K. Petersen, Some Sturmian symbolic dynamics, Available from: http://petersen.web.unc.edu/some-slides-from-talks/

[21]

N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.

[22]

A. Rényi, Representations of real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hungar., 8 (1957), 477-493. doi: 10.1007/BF02020331.

[23]

V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist., 36 (1965), 423-439. doi: 10.1214/aoms/1177700153.

[24]

P. Veerman, Symbolic dynamics of order-preserving orbits, Physica D, 29 (1987), 191-201. doi: 10.1016/0167-2789(87)90055-8.

show all references

References:
[1]

J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.

[2]

V. Anagnostopoulou, Sturmian Measures and Stochastic Dominance in Ergodic Optimization, Ph.D. thesis, Queen Mary University of London, 2009.

[3]

V. Anagnostopoulou and O. Jenkinson, Which beta-shifts have a largest invariant measure?, Jour. Lon. Math. Soc., 79 (2009), 445-464. doi: 10.1112/jlms/jdn070.

[4]

J. Berstel and P. Séébold, Sturmian words, in Algebraic Combinatorics on Words (Encyclopaedia of Mathematics and its Applications 90), (M. Lothaire), Cambridge University Press, (2002), 45–110.

[5]

T. Bousch, Le poisson n'a pas d'arȇtes, Ann. Inst. Henri Poincaré (Proba. et Stat.), 36 (2000), 489-508. doi: 10.1016/S0246-0203(00)00132-1.

[6]

T. Bousch, Une propriété de domination convexe pour les orbites sturmiennes, Can. Jour. Math., 67 (2015), 90-106. doi: 10.4153/CJM-2014-009-8.

[7]

T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, tetris heaps, and the finiteness conjecture, Jour. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2.

[8]

S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Camb. Phil. Soc., 115 (1994), 451-481. doi: 10.1017/S0305004100072236.

[9]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford University Press, 1979.

[10]

O. Jenkinson, Conjugacy Rigidity, Cohomological Triviality, and Barycentres of Invariant Measures, Ph.D. thesis, University of Warwick, 1996.

[11]

O. Jenkinson, Frequency locking on the boundary of the barycentre set, Exp. Math., 9 (2000), 309-317.

[12]

O. Jenkinson, Maximum hitting frequency and fastest mean return time, Nonlinearity, 18 (2005), 2305-2321. doi: 10.1088/0951-7715/18/5/022.

[13]

O. Jenkinson, Ergodic optimization, Discrete & Cont. Dyn. Sys., 15 (2006), 197-224. doi: 10.3934/dcds.2006.15.197.

[14]

O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1-12. doi: 10.1090/S1079-6762-07-00170-9.

[15]

O. Jenkinson, A partial order on $× 2$-invariant measures, Math. Res. Lett., 15 (2008), 893-900. doi: 10.4310/MRL.2008.v15.n5.a6.

[16]

T. KamaeU. Krengel and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Prob., 5 (1977), 899-912.

[17]

H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, 3rd edition, Springer, 2016. doi: 10.1007/978-3-319-21708-6.

[18]

T. Lindvall, On Strassen's theorem on stochastic domination, Electron. Commun. Probab., 4 (1999), 51-59. doi: 10.1214/ECP.v4-1005.

[19]

M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[20]

K. Petersen, Some Sturmian symbolic dynamics, Available from: http://petersen.web.unc.edu/some-slides-from-talks/

[21]

N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.

[22]

A. Rényi, Representations of real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hungar., 8 (1957), 477-493. doi: 10.1007/BF02020331.

[23]

V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist., 36 (1965), 423-439. doi: 10.1214/aoms/1177700153.

[24]

P. Veerman, Symbolic dynamics of order-preserving orbits, Physica D, 29 (1987), 191-201. doi: 10.1016/0167-2789(87)90055-8.

Figure 1.  Orbits supporting measures $\mu_{110010} \prec \mu_{110110010} = \mu_{110}\ast\mu_{110010}\prec \mu_{110}$
Figure 3.  A concatenation procedure that generates the largest point in the support of a Sturmian measure $S_{p/q}$
Figure 2.  The pairwise incomparable shift-invariant probability measures $\mu_{10}$, $\mu_{1100}$, $\mu_{110100}$ and $\mu_{110010}$ of frequency $1/2$.
Figure 4.  Hasse diagram of first-order stochastic dominance for measures supported on periodic orbits of period up to $7$. Orbits that carry measures with equal frequency are displayed on the same horizontal line, and frequencies decrease from top to bottom.
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