Stochastic extrema as stationary phases of characteristic functions

Sto c hastic extrema as stationar y phases of c haracteristic functions Sergey Nikitin ∗ No ve mber 23, 2018 Abstract The pap er is dealing with semi-classical asymptotics of a characteristic function for a stoc hastic pro cess. The main technical tool is pro v ided b y the stationary phase metho d. The extremal range for a sto chastic pro cess is defined by limit va lues of the complex logarithm of the c haracteristic function. The pap er also outlines a numerical method f or c alculating stochastic extrema. 1 INTR ODUCTION The extremum for a stochastic pro c ess admits tra nsparent n umer ical presen- tation in ter ms of limit s et of its c ha racteristic function w hich is trea ted a s a high frequency integral [7], [10], [11], [1 2]. The pr op osed co ncept o f sto chas- tic extr emum is compatible with other kno wn metho ds of assessing extrema o f sto chastic functions (see, e.g., [2], [3], [4], [5], [8], [6 ]). How ever, the ideo logy of high frequency integrals, though differen t, is clos e to the sim ulated a nnealing techn ique [2], [3]. In our approach the role of the parameter like the in verse of ”temp erature” (from the annealing pro c ess ) is played by the fr e quency and in order to calculate th e extrem um w e increase the frequency . Method of high fre- quency integrals is able to ca lculate extrema of sto chastic pro cesses with several v a riables, and therefore can b e applied to analyz e imag e s and three- dimensional data s amples. When it is p ossible to apply the metho d of assess ing extrema with principal component functions (in the sense of Karhunen-Lo` eve repres ent ation) [6], then our appro ach w o r ks as well and practically leads to the same results. How ever, it does not rely on any type of Ka rhunen-Lo ` eve represen tations and in this sense o ur concept of sto chastic extremum is of equal or mo r e gener al nature than the Kar hunen-Lo ` eve repre s entation itself. Moreov er, the metho d of treating sto chastic e x trema as stationary phases of its c hara c ter istic function ∗ Departmen t of M athematics and Statistics, Arizona State Universit y , T empe, AZ 85287- 1804 nikitin@ asu.edu 1 leads us to a trans pa rent numerical pr o cedure that a llows e fficie n t estimation o f the sto chastic extrem um a nd ev a luation of its statistica l significance. 2 ST A TIONAR Y PHA SE Our goal is to introduce the concept of extre mum for a sto chastic pro cess. In order to do that w e employ the high-frequency integrals: I ( k , ω ) = Z ∞ −∞ ϕ ( t, ω ) e ik · f ( t,ω ) dt, where ω ∈ Ω is a parameter ; both f a nd ϕ are real functions that are infinitely many times differentiable with resp ect to t. Mo reov er , ϕ ( t, ω ) ha s a finite time suppo rt for a n y fixed ω ∈ Ω , supp t ( ϕ ) is a subset of a closed in ter v al from R . Throug hout the paper R denotes the set of real n umber s. F or rea der conv e- nience, we recall the basic prop er ties o f the high-fre q uency int e grals (for further reading on this sub ject see, e.g. [10], [11], [12]). If supp t ( ϕ ) ⊂ [ a, b ] and d dt f ( t, ω ) 6 = 0 ∀ t ∈ [ a, b ] and ∀ ω ∈ Ω then I ( k , ω ) = Z ∞ −∞ ϕ ( t, ω ) e ik · f ( t,ω ) dt = Z b a ϕ ( t, ω ) e ik · f ( t,ω ) dt and int e grating by parts n times yields I ( k , ω ) = ( 1 ik ) n Z b a L n ( ϕ )( t, ω ) e ik · f ( t,ω ) dt where the linear o p e rator L is defined a s L ( ϕ ) = − d dt ( ϕ f t ) and f t denotes the deriv ative of f ( t, ω ) with resp ect to time, f t ( t, ω ) = d dt f ( t, ω ) . As one can see, if the phase f ( t, ω ) do es not hav e critical p oints in s u pp t ( ϕ ) then I ( k , ω ) = O ( 1 k n ) ∀ n ∈ N and ∀ ω ∈ Ω , where N denotes the set o f natural num b ers. This fact is often represented as I ( k , ω ) = O ( 1 k ∞ ) ∀ ω ∈ Ω . The cr itical (or stationa r y) points of the phas e f ( t, ω ) mak e the main contribu- tion in to the high- frequency in teg ral I ( k , ω ) as k → ∞ . 2 Definition 1 A p oint ( t ⋆ , ω ⋆ ) ∈ R × Ω is c al le d a stationary phase (p oint) if d dt f ( t ⋆ , ω ⋆ ) = 0 . The set o f all stationar y phase points for f is a ddressed as S t ( f ) ⊂ R × Ω . Now let us turn o ur atten tion to calculating the contribution of a sta tionary phase point ( t ⋆ , ω ⋆ ) ∈ S t ( f ) in to the high-frequency in teg ral I ( k , ω ) . A station- ary pha se p o int ( t ⋆ , ω ⋆ ) ∈ S t ( f ) is sa id to hav e order m ∈ N if m is the first natural num b er for which ( d dt ) m f ( t ⋆ , ω ⋆ ) 6 = 0 . The set of such stationary phase p oints is denoted by S t m ( f ) . The T aylor ex - pansion near t ⋆ is f ( t, ω ⋆ ) − f ( t ⋆ , ω ⋆ ) = 1 m ! ( d dt ) m f ( t ⋆ , ω ⋆ )( t − t ⋆ ) m + O (( t − t ⋆ ) m +1 ) as t → t ⋆ . Consider the change of co o rdinates x ( t, ω ⋆ ) = ( sign ( f ( m ) t ( t ⋆ , ω ⋆ )) · ( f ( t, ω ⋆ ) − f ( t ⋆ , ω ⋆ ))) 1 m , where sig n ( f ( m ) t ( t ⋆ , ω ⋆ )) denotes the sig n o f ( d dt ) m f ( t ⋆ , ω ⋆ ) . Since d dt x ( t, ω ⋆ ) = | 1 m ! ( d dt ) m f ( t ⋆ , ω ⋆ ) | 1 m + O ( t − t ⋆ ) the change of coo rdinates is not degener ate on some interv al Q ε , t ⋆ − ε < t < t ⋆ + ε Let us take infinitely many times differen tia ble function h with supp t ( h ) ⊂ Q ε (the set of such functions is denoted a s C ∞ 0 ( Q ε )) . Assume also that h ( t, ω ⋆ ) = 1 in a neighbor ho o d of t ⋆ . Then I ( k , ω ⋆ ) = Z ∞ −∞ ϕ ( t, ω ⋆ ) h ( t, ω ⋆ ) e ik · f ( t,ω ⋆ ) dt + Z ∞ −∞ ϕ ( t, ω ⋆ )(1 − h ( t, ω ⋆ )) e ik · f ( t,ω ⋆ ) dt and in order to find the c o ntribution of the stationary phase ( t ⋆ , ω ⋆ ) w e need to calculate asymptotics for Z ∞ −∞ ϕ ( t, ω ⋆ ) h ( t, ω ⋆ ) e ik · f ( t,ω ⋆ ) dt. After making the change of co or dina tes x = x ( t, ω ⋆ ) in the integral I ( k , ω ⋆ ) = e ikf ( t ⋆ ,ω ⋆ ) · Z ∞ −∞ ϕ ( t, ω ⋆ ) h ( t, ω ⋆ ) e ik ( f ( t,ω ⋆ ) − f ( t ⋆ ,ω ⋆ )) dt 3 we have I ( k , ω ⋆ ) = e ikf ( t ⋆ ,ω ⋆ ) · Z ∞ −∞ ϕ ( x, ω ⋆ ) h ( x, ω ⋆ ) x t e sign ( f ( m ) t ( t ⋆ ,ω ⋆ )) · ikx m dx, where x t denotes d dt x ( t, ω ⋆ ) . The integral Z ∞ −∞ e ± ikx m dx can b e calculated by reducing it to the linear combination o f the integrals lik e Z ∞ 0 e ± ikx m dx and then ev aluating the latter with the help of the in tegral a lo ng the c ur ve on a complex plane [1 2]. T he curve consists out of the segment o f x -axis 0 ≤ x ≤ ρ, the arc of a cir cle ρ · e ± iτ (0 ≤ τ ≤ π 2 m ) and the segment of the straig ht line r · e ± i π 2 m ( ρ ≥ r ≥ 0) . T aking ρ → ∞ yields that Z ∞ −∞ e ± ikx m dx = co s( π 2 m ) 1 k 1 m · C m for o dd m and Z ∞ −∞ e ± ikx m dx = e ± i π 2 m k 1 m · C m for even m where C m = 2 · Z ∞ 0 e − x m dx. T aking in to account tha t h ( t ⋆ , ω ⋆ ) = 1 , for even m w e ha ve I ( k , ω ⋆ ) = ϕ ( t ⋆ , ω ⋆ ) · C m ·  m ! k · | f ( m ) t ( t ⋆ , ω ⋆ ) |  1 m · e sign ( f ( m ) t ( t ⋆ ,ω ⋆ )) i π 2 m · e ikf ( t ⋆ ,ω ⋆ ) + e ikf ( t ⋆ ,ω ⋆ ) Z ∞ −∞ ( ϕ ( x, ω ⋆ ) · h ( x, ω ⋆ ) x t −  ϕ ( x, ω ⋆ ) h ( x, ω ⋆ ) x t    x =0 ) e sign ( f ( m ) t ( t ⋆ ,ω ⋆ )) · ikx m dx and the latter integral has the as y mptotic O ( 1 k 2 m ) as k → ∞ as long as there a re no other statio nary phase p oints presen t. If m is o dd then I ( k , ω ⋆ ) = ϕ ( t ⋆ , ω ⋆ ) · C m ·  m ! k · | f ( m ) t ( t ⋆ , ω ⋆ ) |  1 m · cos( π 2 m ) · e ikf ( t ⋆ ,ω ⋆ ) + O ( 1 k 2 m ) 4 as k → ∞ . In conclusion to this sectio n we for mulate the bas ic results of stationar y phase metho d in the form of the following formal statement . Theorem 2.1 L et ϕ ( t, ω ) , f ( t, ω ) b e r e al fun ctions that ar e infinitely many times differ ent iable with r esp e ct to t for any ω ∈ Ω . Mor e over, for any ω ∈ Ω one c an find an interval [ a ( ω ) , b ( ω )] ⊂ R such t hat ϕ ( t, ω ) ∈ C ∞ 0 ([ a ( ω ) , b ( ω )]) . Then the fol lowing statements hold. i. If [ a ( ω ) , b ( ω )] ∩ S t ( f ) = ∅ then I ( k , ω ) = O ( 1 k ∞ ) as k → ∞ ii. If [ a ( ω ) , b ( ω )] ∩ S t ( f ) = { t j } then I ( k , ω ) has the asymptotic X even m j ϕ ( t j , ω ) · C m j ·  m j ! k · | f ( m j ) t ( t j , ω ) |  1 m j · e sign ( f ( m j ) t ( t j ,ω )) i π 2 m j · e ikf ( t j ,ω ) + O ( 1 k 2 m j ) ! + X o dd m j ϕ ( t j , ω ) · C m j ·  m j ! k · | f ( m j ) t ( t j , ω ) |  1 m j · cos( π 2 m j ) · e ikf ( t j ,ω ) + O ( 1 k 2 m j ) ! as k → ∞ . 3 STOCHASTIC EXTREMUM Consider a real v alued sto chastic pro cess ξ ( t ) defined in the pr obability space (Ω , A (Ω) , P ) , where A (Ω) is a σ -a lgebra of subsets from Ω a nd P is a measur e of probability . Througho ut the pap er we assume th at ξ ( t ) tak es o nly non-negative real v a lues. In the con text of this paper it is tacitly as s umed that ξ ( t, ω ) = g ( t, ω ) , where ω is a sto chastic v aria ble whic h do es not dep end o n time and g ( t, x ) is a smo oth non-negative function of its arguments. In order to examine whether ξ ( t ) ha s e xtrema on in terv al [ a, b ] ⊂ R we ta ke a function ϕ ε ( t ) ∈ C ∞ 0 ([ a − ε, b + ε ]) suc h that ε > 0 , ϕ ε ( t ) ≥ 0 ∀ t ∈ R a nd ϕ ε ( t ) = 1 ∀ t ∈ [ a, b ] . 5 Then we analyze the asymptotics of the hig h-frequency integral Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω ) . The formal description of the situation when ξ ( t ) do e s not hav e sto chastic ex- trema on [ a, b ] sounds as follows. Definition 2 L et ln( z ) denote a fi x e d br anch of the c omplex lo garithm. Then a r e al-value d sto chastic pr o c ess ξ ( t ) do es n ot have sto chastic extr ema on [ a, b ] if one c an find a p ositive r e al numb er ε such that lim k →∞ Re { 1 ik ln( Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω )) } = 0 . The logical negatio n of this s ta tement descr ib es the interv als where sto chastic extrema fo r ξ ( t ) o ccur. In other words, the interv al [ a, b ] co nt a ins sto chastic extrema if ∀ ε > 0 lim k →∞ Re { 1 ik ln( Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω )) } 6 = 0 . Sometimes it is pos s ible to estimate extremal v alues for ξ ( t ) on [ a, b ] with the help of the following S max ε ([ a, b ]) = lim k →∞ Re  1 ik ln( Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω ))  (1) and S min ε ([ a, b ]) = lim k →∞ Re  1 ik ln( Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω ))  , (2) where lim and lim denote upper and lo wer limits, resp ectively . F or ma jority of applica tions S max ε ([ a, b ]) , S min ε ([ a, b ]) deliver b oundarie s for extr e mal v a lues of a real stochastic pro cess ξ ( t ) . O ne can justify that when the sto chastic pro cess ξ ( t ) has a dditional pro per ties like, for ex a mple, Ω is a smo oth manifold and ξ ( t, ω ) is a s mo oth function of t and ω . Le t dP dω denote the pro bability densit y function for P ( ω ) . Then the following statemen t takes place. Theorem 3.1 L et Ω b e a smo oth manifol d. Assu me also that density dP dω ex- ists; dP dω ∈ C ∞ (Ω) , ξ ( t, ω ) ∈ C ∞ ( R , Ω ) and ther e is only one stationary phase p oint ( t ⋆ , ω ⋆ ) ∈ S t 2 ( ξ ) ∩ ( [ a − ε, b + ε ] × Ω) with ∂ ∂ ω ξ ( t ⋆ , ω ⋆ ) = 0 and the se c ond derivative ∂ 2 ξ ∂ t∂ ω has non-zer o determinant J ( t ⋆ , ω ⋆ ) = det  ∂ 2 ξ ∂ t∂ ω ( t ⋆ , ω ⋆ )  6 = 0 (3) 6 at the stationary phase p oint ( t ⋆ , ω ⋆ ) . If d dω P ( ω ⋆ ) > 0 then the fol lowing is true: Re  1 ik · ln  Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω )  = ξ ( t ⋆ , ω ⋆ )+ O ( 1 k ) as k → ∞ . Pro of. Let n denote the dimension of Ω . Then applying the stationar y phase metho d [11] ,[12] to Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t,ω ) dtdP ( ω ) yields Z Ω Z ∞ −∞ ϕ ε ( t ) e ikξ ( t ) dtdP ( ω ) = d dω P ( ω ⋆ ) · ( 2 π k ) n +1 2 · 1 p | J ( t ⋆ , ω ⋆ ) | · e sign ( ∂ 2 ξ ∂ t∂ ω ( t ⋆ ,ω ⋆ )) i π 4 · e ikξ ( t ⋆ ,ω ⋆ ) + O ( 1 k 1+ n +1 2 ) where sig n ( ∂ 2 ξ ∂ t∂ ω ( t ⋆ , ω ⋆ )) denotes the difference betw ee n the num b er o f p ositive and the n umber of neg- ative eigenv alues of the co rresp onding quadra tic form. T ak ing the co mplex logarithm of d dω P ( ω ⋆ )( 2 π k ) n +1 2 · 1 p | J ( t ⋆ , ω ⋆ ) | · e sign ( ∂ 2 ξ ∂ t∂ ω ( t ⋆ ,ω ⋆ )) i π 4 · e ikξ ( t ⋆ ,ω ⋆ ) · (1 + O ( 1 k )) we o btain ln( Z ∞ −∞ ϕ ε ( t ) e ikξ ( t ) dt ) = i k ξ ( t ⋆ , ω ⋆ ) − n + 1 2 ln( k )+ln( d dω P ( ω ⋆ ))+ n + 1 2 · ln(2 π ) − 1 2 ln  | J ( t ⋆ , ω ⋆ ) |  + i π 4 · si g n ( ∂ 2 ξ ∂ t∂ ω ( t ⋆ , ω ⋆ )) + O ( 1 k ) as k → ∞ . Dividing by ik and taking rea l part complete the pro o f. Q.E.D. F ormulas (1), (2) together with Theor em 3 .1 give us a recip e for ass essing extrema o f a real sto chastic pro ces s ξ ( t ) on an arbitrar y time interv al [ a, b ] . In order to do that one needs to analyze the limit prop erties of the following int e g ral for big v alues o f k : Re  1 ik Z γ k dz z  , where the complex cur ve γ k is defined a s γ k =  Z Ω Z ∞ −∞ ϕ ε ( t ) e iλξ ( t,ω ) dtdP ( ω ); 0 ≤ λ ≤ k  . 7 Notice that γ k can b e in terpre ted as the characteristic function o f ξ ( t, ω ) , where ϕ ε ( t ) is chosen so that Z ∞ −∞ ϕ ε ( t ) dt = 1 . In conclusion we notice that the condition (3) can be relaxed with the help of Theorem 2 .1 (see also [1], [9]). In some applications (e.g., when estimating tails of distributions ) it is b eneficial that the truncation function ϕ ε ( t ) also depends on ω . References [1] A tiyah M.F., ”Res olution of singularities a nd division of distributio ns ”, Comm. Pur e Appl. Math. , vol.23, No. 2, 1970, pp.145-1 5 0. [2] Bo chac hevsk y I.O., Johnson M.E., Stein M.L.” Generalized simulated an- nealing for function o ptimization”, T echnometrics, 28 (1986), pp. 209-2 17. [3] Bro oks S.P ., Morgan B.J.T., ” Optimization using simulated annealing”, The Statistician , vol.44, No.2 (1995), pp. 241-26 7. [4] Burry K.V., S tatistic al Metho ds in Appli e d Scienc e, John Wiley & Sons , 1975. [5] Drees H., de Ha an L., Li D., ”Approximations to the tail empirical distribu- tion function with applicatio n to testing extr eme v a lue conditions”, Journ al of Statistic al Planning and Infer enc e , v ol. 1 36, issue 10, (1 O ctob er 20 06), pp. 3498 -353 8 . [6] Hall P ., Vial C . ”Assessing extrema of empir ic a l principal co mpo nent func- tions”, The Annals of Statistics, 200 6 , v ol. 34, No. 3, pp.1518-1 5 44. [7] Guillemin V., Sternberg S., Ge ometric asymptotics, Mathematical Surveys, Num. 14, American Mathematical So ciety , Providence, Rho de Isla nd, 1977 . [8] Gum b el, E.J ., ”Les v aleurs extrmes des distributio ns statistiques,” Ann. Inst. H. Poinc ar , 5, (1935), pp. 115- 158. [9] Malgrang e B., ” Int e g rales asymptotiques et mono dromie” , Ann. S ci. Ec ole Norm. Sup er. , vol.7, 1974, pp.405-43 0. [10] McClur e J.P ., W ong R., ”J ustification of the Stationary Phase Approxima- tion in Time-Domain Asy mptotics”, Pr o ceedings: Mathematica l, P hysical and E ngineering Sciences , V ol. 453, No. 1960 (May 8 , 1997), pp. 1019- 1 031 [11] V.P . Ma slov, The c omplex WKB metho d for nonline ar e quations I , Birkhauser V erlag, Ba sel, Boston, Berlin, 199 4. [12] V.P . Maslov and F edoriuk, Semi-classic al appr oximation in quantum me- chanics , D.Reidel Dordrecht , Holla nd, 1981. 8