A Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone

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1 mathmatics Articl A Simpl Formula for th Hilbrt Mtric with Rspct to a Sub-Gaussian Con Stéphan Chrétin 1, * and Juan-Pablo Ortga 2 1 National Physical Laboratory, Hampton Road, Tddinton TW11 0LW, UK 2 Faculty of Mathmatics and Statistics, Univrsity of St. Galln, CH-9000 St. Galln, Switzrland; juan-pablo.ortga@unisg.ch * Corrspondnc: stphan.chrtin@npl.co.uk Rcivd: 11 January 2018; Accptd: 19 Fbruary 2018; Publishd: 2 March 2018 Abstract: Th Hilbrt mtric is a widly usd tool for analysing th convrgnc of Markov procsss and th rgodic proprtis of dtrministic dynamical systms. A usful rprsntation formula for th Hilbrt mtric was givn by Livrani. Th goal of th prsnt papr is to xtnd this formula to th non-compact and multidimnsional stting with a diffrnt con, taylord for sub-gaussian tails. Kywords: dynamical systms; Hilbrt mtric; Livrani s formula 1. Introduction Lt V b a topological vctor spac and C a closd convx con insid V njoying th proprty that C C =. Thn, C dfins a partial ordring by stting Using this ordring, th Hilbrt smi-mtric Θ on C is givn by f g g f C 0}. 1) Θ f, g) = log [ ] β f,g), 2) α f,g) whr and α f, g) = supλ R + λ f g}, 3) β f, g) = µ R + g µ f } 4) and α = 0 and β = + if th corrsponding sts ar mpty. In th squl, w will focus on th cas whr V is th spac of continuous intgrabl functions on R n with Euclidan norm dnotd by. Th Hilbrt mtric, originally introducd in [1], has bn vry usful in rgodic thory for dtrministic dynamical systms [2 4], for th analysis and application of Markov Chains to control systms statistics and ormation thory [5 7]. Robust routing problms hav also bn studid using th Hilbrt mtric viw point [8]. Th first application of th Hilbrt mtric to th fild of dynamical systms is Birkhoff s approach to th Prron-Frobnius thorm [9]; s also [10]. Th ground-braking rsult of Birkhoff is th following. Thorm 1. Lt C b a con in a vctor spac V and K b a con in a vctor spac W. If L : V W is a linar mapping with LC) K and projctiv diamtr L) = sup x, y C, Lx) K Ly) dlx, Ly). Thn ) 1 dlx, Ly)/dx, y) tanh x C y C 4 L) 5) Mathmatics 2018, 6, 35; doi: /math

2 Mathmatics 2018, 6, 35 2 of 5 whr tanh ) = 1. Birkhoff s thorm provids an lgant way to prov that crtain maps btwn cons ar contracts and thrfor obtain xistnc and uniqunss for crtain problms such as in Prron-Frobnius thory for positiv oprators. Th goal of this short not is to xtnd to th noncompact and multidimnsional stting a usful formula for th Hilbrt smi-mtric which was prviously givn by Livrani [2] in th cas in which V is th spac L 1 [0, 1]) of intgrabl functions of th intrval [0, 1]. Th con C in that rsult, which will b dnotd in th squl by C a, a > 0, is givn by C a [0, 1]) = g C 0 [0, 1]) L 1 [0, 1]) x, y [0, 1] g > 0 and gx) gy) a x y }. 6) In that stup, th Hilbrt smi-mtric is givn by th following xprssion: 2. Main Rsult [ Θ f, g) = ln sup x,y,u,v [0,1] a x y 2 gy) gx) ) a u v 2 f v) f u) ) a x y 2 f y) f x) ) a u v 2 gv) gu) ) ]. 7) Th con w us in th squl is diffrnt from th con chosn by Livrani in [1]. C a R n ) = g C 0 R n ) L 1 R n ) x, y R n g > 0 and gx) gy) a x y 2} 8) This con is chosn to contain dnsitis with subgaussian tails on R n. Our main rsult is th following thorm. Thorm 2. Lt Θ b th Hilbrt smi-mtric associatd to C a, a > 0; whn f, g C a, [ Θ f, g) = ln sup x,y,u,v E a x y 2 gy) gx) ) a u v 2 f v) f u) ) a x y 2 f y) f x) ) a u v 2 gv) gu) ) ]. 9) Proof. W hav to comput α f, g) and β f, g). Just as in th proof p. 247 of Lmma 2.2 of [2], w obtain gx) α f, g) = min x R n f x) ; a x y 2 gy) gx) x, y R n a x y 2 f y) f x) }. 10) W now prov that gx) x R n f x) a x y 2 gy) gx) x, y R n a x y 2 f y) f x) 11) Lt x n ) n N R n b a minimizing squnc for th lft hand sid of Equation 11). W hav to split th analysis into two cass. First cas. Assum that x n ) n N has a boundd subsqunc, dnotd by x σn) ) n N. Morovr, by th Bolzano-Wirstrass Thorm w may assum without loss of gnrality that this subsqunc convrgs to som limit point x. Fix ɛ > 0 and lt N N b such that gx σn) ) f x σn) ) x R n gx) f x) + ɛ 12)

3 Mathmatics 2018, 6, 35 3 of 5 for all n N. Now tak x such that Thn, w gt sup a x x σn) 2 f x σn) ) f x) < +. 13) n N a x x σn) 2 gx σn) ) gx) a x x σn) 2 f x σn) ) f x) Lt n tnd towards +, and obtain = a x x σn) 2 gx σn) ) f x σn) ) f x σn)) gx) f x) f x) a x x σn) 2 f x σn) ) f x) a x x σn) 2 gx) f x) f x σn)) + ɛ ) gx) f x) f x) a x x σn) 2 f x σn) ) f x) = gx) f x) + ɛ a x xσn) 2 f xσn) ) a x x σn) 2 f x σn) ) f x). 14) a x x 2 gx ) gx) a x x 2 f x ) f x) gx) f x) + ɛ a x x 2 f x ) a x x 2 f x ) f x) 15) and th rsult follows by taking ɛ 0 du to th assumption of Equation 13) on x. Th rsult is thn asily sn to hold for all x by continuity. Scond cas. In this cas, x n x +. Considr th unit vctor d n givn by d n = x n x x n x 16) and xtract a convrgnt subsqunc d σn) ) n N dnoting its limit by d. Now tak z δ = x δd. Lt us prov that a z δ x σn) 2 gx σn) ) gz δ ) lim n + a z δ x σn) 2 f x σn) ) f z δ ) gz δ) f z δ ) Ltting δ tnd towards zro will prov th dsird rsult using continuity. As in th first cas, on asily finds that for ach ɛ > 0 thr xists N N such that for all n N, 17) a z δ x σn) 2 gx σn) ) gz δ ) a z δ x σn) 2 f x σn) ) f z δ ) gz δ) f z δ ) + ɛ 1 1 f z δ ) a z δ x σn) 2 f x σn) ) = gz δ) f z δ ) + ɛ 1 f x) 1 f z δ ) a x x σn) 2 a x x σn) 2 f x σn) ) a z δ x σn) 2 f x σn) ). 18) Looking at this xprssion, w obviously wondr about th asymptotic bhavior of c n = f x) f z δ ) a x x σn) 2 a x x σn) 2 f x σn) ) a z δ x σn) 2 f x σn) ). 19) For this purpos, first notic that sinc f blongs to C a, w hav f x) a x x 1. 20) σn) 2 f x σn) )

4 Mathmatics 2018, 6, 35 4 of 5 and but f z δ ) a x x σn) 2 a z δ x σn) 2 f x σn) ) a x x σn) 2 a x z δ 2 a z 21) δ x σn) 2 a x x σn) 2 a z δ x σn) 2 +a x z δ 2 = 2a x x σn) x z δ, 22) and sinc x n x +, w obtain that c n 0 as n +, implying that th valu on is not an accumulation point. Using this, w may dduc from Equation 18) that Ltting ɛ tnd towards zro w obtain a z δ x σn) 2 gx σn) ) gz δ ) a z δ x σn) 2 f x σn) ) f z δ ) gz δ) + ɛ. 23) f z δ ) a z δ x σn) 2 gx σn) ) gz δ ) n N a z δ x σn) 2 f x σn) ) f z δ ) gz δ) f z δ ). 24) Ltting δ tnd towards zro and using continuity, w finaly conclud that a x x σn) 2 gx σn) ) gx) n N a x x σn) 2 f x σn) ) f x) gx) f x), 25) implying th dsird rsult. 3. Conclusions In this short not, w prsntd an xtnsion of a formula for th Hilbrt mtric obtaind in [2] to th multidimnsional and non-compact cas for a diffrnt rfrnc con. Using th prsntd formula and Thorm 1, applications to th study of Markov Chain Mont Carlo mthods and th analysis of th Sinkhorn algorithms for Optimal Transportation will b undrtakn in futur work. Author Contributions: Stéphan Chrétin and Juan-Pablo Ortga workd qually on th contnt and on th writing of th papr. Conflicts of Intrst: Th authors dclar no conflicts of intrst. Rfrncs 1. Hilbrt, D. Übr di grad Lini als kürzst Vrbindung zwir Punkt. Math. Ann. 1895, 46, Livrani, C. Dcay of corrlations. Ann. Math. 1995, 142, Baladi, V. Positiv Transfr Oprators and Dcay of Corrlations; Advancd Sris in Nonlinar Dynamics, 16; World Scintific Publishing Co., Inc.: Rivr Edg, NJ, USA, Naud, F. Birkhoff cons, symbolic dynamics and spctrum of transfr oprators. Discrt. Contin. Dyn. Syst. 2004, 11, Bodnart, R.; Stttnr, L. Asymptotics of controlld finit mmory filtrs. Syst. Control Ltt. 2002, 47, Di Masi, G.B.; Stttnr, L. Ergodicity of hiddn Markov modls. Math. Control Signals Syst. 2005, 17, L Gland, F.; Oudjan, N. Stability and uniform approximation of nonlinar filtrs using th Hilbrt mtric and application to particl filtrs. Ann. Appl. Probab. 2004, 14, Chn, Y.; Gorgiou, T.; Pavon, M.; Tannnbaum, A. Rlaxd Schrodingr bridgs and robust ntwork routing. arxiv 2018, arxiv:

5 Mathmatics 2018, 6, 35 5 of 5 9. Birkhoff, G. Extnsions of Jntzsch s thorm. Trans. Am. Math. Soc. 1957, 85, Kohlbrg, E.; Pratt, J. Th contraction mapping approach to th Prron-Frobnius thory: Why Hilbrt s mtric? Math. Opr. Rs. 1982, 7, c 2018 by th authors. Licns MDPI, Basl, Switzrland. This articl is an opn accss articl distributd undr th trms and conditions of th Crativ Commons Attribution CC BY) licns