ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES OF THE SPACE Sub ϕ (Ω)

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1 Teo Imov.taMatem.Statist. Theo. Pobability and Math. Statist. Vip. 7, 4 No. 7, 5, Pages 5 7 S Aticle electonically published on Decembe 8, 5 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES OF THE SPACE Sub ϕ Ω UDC 59. YU. V. KOZACHENKO AND I. V. ROZORA Abstact. Stochastic pocesses of the space Sub ϕ Ω ae consideed in the pape. We pove uppe bounds fo lage deviation pobabilities and constuct models of stochastic pocesses in the space C[, ] with a given accuacy and eliability. Stongly sub-gaussian pocesses ae also consideed as a paticula case.. Intoduction We conside stationay stochastic pocesses of the space Sub ϕ Ω and constuct models of these pocesses with a given accuacy and eliability. Simila poblems fo models of some stochastic pocesses ae consideed in [] [3]. In the poofs below we follow the method of []. Section of this pape contains basic definitions and popeties of stochastic pocesses of the space Sub ϕ Ω. Moe detail can be found in [4]. Stationay stochastic pocesses with discete spectum ae consideed in Section 3. Section 4 is devoted to models of stationay pocesses of the space Sub ϕ Ω. We obtain esults fo models appoximating a stochastic pocess with a given accuacy and eliability in the Banach space C[, ]. A paticula case of sub-gaussian pocesses is consideed in Section 5.. Stochastic pocesses of the space Sub ϕ Ω Definition. [4]. A convex even continuous function ϕx such that ϕ = is called an N-function if ϕx > fox,ϕx/x asx, and ϕx/x as x. Lemma. [4]. Afunctionϕx, x R, isann-function if and only if ϕx = x lu du, x R, whee the density lu, u, is a nondeceasing ight continuous function such that l =, lx > fo x,andlx as x. Definition.. Let fx, x R, be a eal function. A function f x, x R, is called the Young Fenchel tansfom of the function f o the conjugate function to f if f x =supxy fy. y R Mathematics Subject Classification. Pimay 68U; Seconday 6G. Suppoted in pat by NATO gant PST.CLG c 5 Ameican Mathematical Society

2 6 YU. V. KOZACHENKO AND I. V. ROZORA Lemma. [4]. If fx, x R, isann-function, then f x, x R, isalsoan N-function. Moeove f x =xy fy fo all x>, wheey = l x and l x is the genealized invese function fo the density lx. Now we give the definition of the space Sub ϕ Ω and stochastic pocesses of the space Sub ϕ Ω. Let ϕx beann-function and let thee exist constants x and c > such that ϕx =c x fo x <x. By Ω,B,P we denote the standad pobability space. Definition.3. The space of centeed andom vaiables ξ such that fo all λ R thee exists a constant fowhich E expλξ} expϕλ} is called the space Sub ϕ Ω of andom vaiables. Theoem. [4]. The space Sub ϕ Ω is a Banach space with espect to the nom ϕ ln E expλξ} τ ϕ ξ =sup λ> λ whee ϕ is the genealized invese function to ϕ. Then the nom τ ϕ ξ is such that E expλξ} expϕλτ ϕ ξ} fo all λ R. Moeove thee exists a constant c> such that E ξ / <cτϕ ξ. Theoem. [4]. Let ξ,ξ,...,ξ n be independent andom vaiables of the space Sub ϕ Ω. Ifϕ x is a convex function, then n n ξ i τϕξ i. τ ϕ i= If ϕx =x /, then andom vaiables ξ of the space Sub ϕ Ω ae called sub-gaussian. We denote the space of sub-gaussian andom vaiables by Sub x /Ω = SubΩ. If Eξ = τ ξ, then the andom vaiable ξ is called stongly sub-gaussian. The family of all stongly sub-gaussian andom vaiables defined on the standad pobability space is denoted by SSubΩ. Definition.4. We say that a stochastic pocess ξt, t [,T], belongs to the space Sub ϕ Ω if ξt Sub ϕ Ω fo any fixed t [,T]andsup t [,T ] τ ϕ ξt <. In the space Sub ϕ Ω equipped with the nom τ ϕ, conside a stochastic pocess ξ = ξt,t [, ]}, whee ϕx isann-function such that ϕ x is convex. The following esult is poved in [5]. i=

3 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES 7 Lemma.3. Let ξ = ξt, t [, ]} be a sepaable stochastic pocess of the space Sub ϕ Ω. Assume that thee exists an inceasing continuous function σh, h, such that σh as h and sup τ ϕ ξt ξs <σh. t s <h Let γ =sup t [,] τ ϕ ξt, β σ,andletu,u } be a nondeceasing continuous function such that u fo u, =, andthefunctionexpu} is convex. Assume that β θ ϕ u du <, whee N σ u θ ϕ u = ϕ ln N σ u, Nu is the metic massiveness, that is, the minimal numbe of closed balls of adius u that cove the inteval [, ]. Then E exp λ sup ξt t [,] fo all λ R and p,. } exp We pove the following esult. λγ ϕ p λγ θ ϕ pβ+ } λβ p+ϕ p p λ βp θ ϕ u du pp Theoem.3. Let the assumptions of Lemma.3 hold and let β minγ,σ }. If x> is such that γ <x, then P sup ξt >x t [,] exp } βγ <xσ } x ϕ γ x l x/γ βγ expϕ} βγ /x, du σ u +. Poof. We apply Lemma.3. The assumption of Theoem.3 implies that β minγ,σ } γ.thus λγ λβ λγ ϕ p+ϕ p ϕ, p p p whence we obtain by the Chebyshev inequality and that 3 P sup ξt >x t [,] } exp λx + ϕ λγ p } I,

4 8 YU. V. KOZACHENKO AND I. V. ROZORA whee I = λ βp λγ θ ϕ pβ+ θ ϕ u du. pp It follows fom Lemma. that xy = ϕx+ϕ y fox = l y, whee ϕ y isthe Young Fenchel tansfom and l y is the invese function to the density lx. Using this esult, we obtain that λγ 4 λx ϕ p = λγ x p λγ x p ϕ = ϕ p γ p γ fo λγ / p =l x p/γ. Thus 5 λ = p x p l. γ γ Note that the function N σ u θ ϕ u = ϕ ln N, u, σ u deceases in u. Hence βp θ ϕ pβ θ ϕ u du βp p βp and λγ βp λγ θ ϕ pβ θ ϕ u du. βp p βp Since γ /β, we have λ βp 6 λγ θ ϕ pβ+ θ ϕ u du γ λ βp θ ϕ u du. p p β p p Now we apply 6 fo I with λ definedin5andget l 7 I / x p/γ βp θ ϕ u du. βp Note that Nu /u +, whence θ ϕ u = Since the function N σ u ϕ ln N σ u deceases in u, u,βp, and βγ <xσ σ u +. σ u + ϕ ln ϕ ln σ u + expϕ}, we obtain, by the assumptions of the theoem, that ϕ ln σ u + >

5 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES 9 fo u,βp. Thus βp θ ϕ u du βp σ u + ϕ ln σ u + βp The latte elation togethe with 3, 4, and 7 implies that 8 } P sup ξt >x t [,] x p exp ϕ γ } l x p/γ βp βp σ u + du. du σ u +. Since p, is abitay, one can substitute p = γ /x < in 8. This will complete the poof of the theoem. Example.. If ϕx =x /, that is, if we deal with the space of sub-gaussian andom vaiables, then the Young Fenchel tansfom and the density fo ϕ ae given by ϕ y = y /andlx =x, espectively. Theoem.3 implies the following esult. Theoem.4. Let ξ = ξt,t [, ]} be a sepaable stochastic pocess of the space SubΩ of sub-gaussian andom vaiables; let a function σh satisfy the assumptions of Lemma.3, γ =sup t [,] τξt, β =minσ,γ }; andletu,u } be a nondeceasing continuous function satisfying all the assumptions of Lemma.3. If x> is such that γ <x, βγ <xσ exp/} then } P sup ξt >x exp } x t [,] γ x x γ βγ βγ /x, du σ u Stationay pocesses with discete spectum Let ξt,t [, ]} be a stationay stochastic pocess such that Eξt =,t [, ], and Eξt + hξt = Bh. Definition 3.. A stationay pocess ξt is called a pocess with discete spectum if its coelation function Bh can be epesented in the fom Bh = b k cos λ k h, k= whee b k >, k= b k <, andλ k ae some numbes such that λ k λ k+ and λ k as k.

6 YU. V. KOZACHENKO AND I. V. ROZORA The latte definition and Kahunen theoem imply that a stationay stochastic pocess ξ = ξt,t [, ]} with discete spectum can be epesented in the fom of the seies 9 ξt = ξ k b k cos λ k t + η k b k sin λ k t, k= whee ξ k and η k ae independent andom vaiables such that Eξ k = Eη k = Eξ k η l =, Eξ k ξ l = Eη k η l = l k, k =,,..., l =,,..., and seies 9 conveges in the mean squae sense. If the andom vaiables ξ k and η k, k =,,..., belong to the space Sub ϕ Ω, then ξt Sub ϕ Ω, too. Similaly, if ξ k and η k, k =,,..., ae stongly sub-gaussian, then ξt SSubΩ. 4. Models of stochastic pocesses of the space Sub ϕ Ω We constuct a model ξ N t of a pocess ξt such that ξ N t appoximates ξt with a given accuacy and eliability in the Banach space C[, ]. Conside a stationay pocess ξ with a discete spectum and assume that the pocess belongs to the space Sub ϕ Ω. In what follows we assume that τ ϕ ξ k =τ ϕ η k =d>. Let the numbes b k be unknown, but we know thei appoximate values b k. We also know that b k b k γ k fo some known constants γ k. Definition 4.. A pocess ξ N t is called a model of ξt if ξ N t = N bk ξ k cos λ k t + η k sin λ k t, k= whee the numbes b k satisfy inequality and ξ k and η k ae independent andom vaiables of the space Sub ϕ Ω such that Eξ k = Eη k = Eξ k η l =, Eξ k ξ l = Eη k η l = l k, k =,...,N, l =,...,N. Definition 4.. We say that a model ξ N t appoximates a pocess ξt withagiven eliability ν, ν,, and accuacy >inthespacec[, ] if } P sup ξt ξ N t > ν. t [,] It is easy to see that the eo of the model is t, N =ξt ξ N t N = b k b k ξ k cos λ k t + η k sin λ k t+ k= := t, N+ t, N. k=n+ b k ξ k cos λ k t + η k sin λ k t Next we find uppe estimates fo τ ϕ t, N, t [, ], and τ ϕ t, N s, N, t, s [, ].

7 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES Theoem. implies that τ ϕ t, N τ ϕ t, N + τ ϕ t, N and τ ϕ t, N N b k b τ k ϕ ξ k cos λ k t + τϕη k sin λ k t k= d N k= γ k := A N. Recall that we conside the case whee τ ϕ ξ k =τ ϕ η k =d>. Similaly to the case of we estimate τϕ t, N d b k := B N. k=n+ To estimate τ ϕ t, N s, N, we use the following auxiliay esult poved in [3]. Lemma 4.. Let ψu, u, be a continuous inceasing function such that ψ =. Assume that the function u/ψu is nondeceasing fo u>u whee u is some constant. Then sin u v ψu + u ψv + u fo all u and v>. Example 4.. The function ψu =u α, α, ], satisfies the assumptions of Lemma 4. fo u =. Thus sin u v uα v α fo u, v >. Example 4.. Anothe example fo Lemma 4. is the function ψu =ln α u+, α>, and u = e α. We have in this case sin u lne α α v + u lne α. + v Using Lemma 4., we get the estimate τϕ t, N s, N 3 d d d N k= N k= b k b k [ cos λ k t cos λ k s +sinλ k t sin λ k s ] γ k cos λ kt s = 4d N γk k= ψ λ k /+u ψ t s + u := C N ψ t s + u. N γk k= sin λ kt s Assume that k= b k ψ λ k /+u <. Following the same idea, one can pove that τϕ t, N s, N d b ψ λ k /+u k ψ t s + u 4 k=n+ := D N ψ t s + u.

8 YU. V. KOZACHENKO AND I. V. ROZORA Theefoe elations 4 and Theoem. imply that 5 τϕ t, N A N + B N, 6 τϕ t, N s, N CN + D N ψ t s + u. In what follows we assume that the function ψu in Lemma 4. is such that ψu as u. Then it follows fom 5 and 6 that 7 γ = CN + D N A N + B N := γ N, σh = ψ /h + u := LN ψ /h + u = σ Nh, whee γ and σh satisfy all the assumptions of Lemma.3. Theoem 4.. Let k= b k ψ λ k /+u <, let the function ψu satisfy the assumptions of Lemma 4., and moeove let ψu as u. Let thee exist a nondeceasing continuous function u,u } such that u fo u, =, and expu} is convex. We also assume that β θu du <, whee θu =ψ LN/u. A stochastic pocess ξ N t is a model appoximating a sepaable pocess ξt of the space Sub ϕ Ω with a given eliability ν, ν,, and accuacy > in the space C[, ] if N is such that γ N <, 8 9 exp β N γ N < } ϕ γ N LN ψ expϕ} + u, l /γ N β N γ N βn γ N/ θu du <ν, whee β N =minγ N,LN/ψ + u } and constants γ N and LN ae defined in 7. Poof. Since ξt, ξ N t Sub ϕ Ω, thei diffeence ξt ξ N t isalsoapocessofthe space Sub ϕ Ω. We poved above that γ = γ N = A N + B N, σh =σ N h =LN ψ /h + u, whence σ u = ψ. LN/u u Thus LN LN σ u + = ψ u + u ψ +. u Now we pove that ψ LN/u > fo all u of the inteval,β N γ N/. Since the function ψ LN/u deceases in u and γ N/ < by the assumptions of the theoem, it emains to pove that ψ LN/β N >.

9 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES 3 The latte inequality is equivalent to the elation LN/ψ >β N that is obviously tue, since } β N =min γ N,σ N σ N LN/ψ. Hence σ u + It is easy to see that LN ψ. u LN σ = expϕ} ψ expϕ} + u. Substituting the uppe bound obtained above into the inequality of Theoem.3, we complete the poof of Theoem 4.. Conside the case of the function ψu =u α, α, ], and u =. Inthiscasewe obtain the following esult. Theoem 4.. Let k= b k λα k <, α, ]. A stochastic pocess ξ N t is a model appoximating a sepaable pocess ξt Sub ϕ Ω with a given eliability ν, ν,, and accuacy > in the space C[, ] if N is such that γ N <, LN β N γ N < α expϕ} α, } b/α exp ϕ γ N α LN l α b β N γ N γ N + <ν, /b whee γ N and LN ae defined in 7, β N =minγ N,LN/ α },and <b<α. Poof. Recall that we deal with the case of ψu =u α, α, ], and u =. Then the inequalities of Theoem 4. imply the fist two inequalities of Theoem 4.. It emains to pove that the thid inequality of Theoem 4. holds. We apply elation 9 with u =u b, b,α. Note that this function u satisfies all the assumptions of Theoem 4. and that ψ u =u /α, α, ]. We have βn γ N/ βn γ N/ b LN θu du = ψ du u βn γ N/ LN b/α = du u = αlnb/α α b b/α βn γ N β N γ N.

10 4 YU. V. KOZACHENKO AND I. V. ROZORA Since u =u + /b fo b,α, we conclude that l /γ N βn γ N/ θu du β N γ N b/α /b α LN = l + α b β N γ N γ N. Substituting the ight-hand side of the latte equality into 9, we complete the poof of the theoem. 5. Models of sub-gaussian stochastic pocesses We conside sub-gaussian stochastic pocesses in this section; that is, we deal with the function ϕu =u /. Then the Young Fenchel tansfom and the density fo ϕ ae ϕ x =x /andlx =x, espectively. The following esult follows fom Theoem 4. Theoem 5.. Let the assumptions of Theoem 4. hold. A model ξ N t appoximates apocessξt with a given eliability ν, ν,, and accuacy > in the space C[, ] if N is such that exp γ N γ N <, } γ N β N γ N βn γ N/ θu du <ν. Poof. Theoem 5. follows fom Theoem 4.. We pove only that elation implies inequality 8 in the case of sub-gaussian pocesses; that is, we pove that γ N < = β N γ N < LN ψ exp/} + u. Indeed, LN ψ exp/} + u = σ σ.7. exp/} Note that β N = minγ N,σ/} σ/. The function σu isinceasingby definition. Thus β N LN / ψ exp/} + u. The theoem is poved. If ψu =u α, α, ], and u =, then Theoems 4. and 5. yield the following esult. Theoem 5.. Let k= b kλ α k <, α, ]. A stochastic pocess ξ N t is a model appoximating a sepaable sub-gaussian pocess ξt with a given eliability ν, ν,, and accuacy > in the space C[, ] if N is

11 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES 5 such that γ N <, exp } γ N b/α /b α LN + α b β N γ N γ N <ν fo some b,α, α, ], wheeβ =minγ N,LN/ α } and γ N and LN ae defined in 7. Example 5.. Conside the case of ψu = u α, α, ], and u =. Let the pocess ξt in 9 be sub-gaussian. This means that the andom vaiables ξ k and η k, k, ae stongly sub-gaussian andom vaiables; that is, d = τ ξ k =Eξk, d = τ η k =Eηk. Thus d =. Let b k = k a, a>, λ k = k, and let the eos of appoximation of the numbes b k be the constants γ k = γ fo all k. Then b kλ k = k a < k= k= and the numbes A N, B N,andγ N = A N + B N ae such that B N = d b k=n+ k = k=n+ A N = d k a = N k= k k=n+ γ k =N +γ, k dx ka N x a dx = a N a, γn N +γ + a N a. Now we find the point of minimum and the minimum value of the ight-hand side of the latte inequality ove all eal numbes N>. It is obvious that N min = /a γ is the point of minimum in this case. Then min γ N = γ /a + γ + / 4a a γ /a γ /a a := γ γ, a >. To constuct the model, we take N =[N min ]+, whee [c] stands fo the intege pat of anumbec. Theoem 5. implies that γ N satisfies condition. All possible γ<inthis case ae detemined by the condition 4a γ /a <= γ< a a/a a. 4a

12 6 YU. V. KOZACHENKO AND I. V. ROZORA Thus a/a a 3 γ, min,. 4a Remak. When we apply Theoem 4., the numbes, ν, a, α, andb,αaswellasγ in 3 ae known. Thus the poblem of constucting a model of a pocess is educed to the poblem of finding the minimal numbe N that satisfies conditions. Now we evaluate LN = C N + D N fo b k = k a, a>, λ k = k,andγ k = γ. Fist, N N k+ C N = α d γkλ α k = α γ k α dx α γ N + α+, α + D N = α d k= k=n+ b kλ α k = α k= k k=n+ Then γ L N α N + α+ + α + Substituting N min into LN, we get +γ LN min = α γ α /a /a α+ γ + α + Thus Theoem 5. can be ewitten as follows. k a α α a α N a α. a α N a α. Theoem 5.3. A stochastic pocess N ξ N t = ξ k cos kt + η k sin kt, a >, N = bk k= / := Lγ. a α [ ] /a +, γ is a model appoximating a sepaable pocess ξt = k a ξ k cos kt + η k sin kt, k a b k <γ, k N, k= with a given eliability ν, ν,, and accuacy > in the space C[, ] if fo b,α, α, ], theeexistsanumbeγ in the inteval a/a a, min, 4a such that exp } b/α /b γ γ α Lγ + α b β γ γ N γ γ <ν, whee β γ =minγ γ,lγ/ α } and 4a γ γ =γ /a a, +γ Lγ = α γ α /a /a α+ / γ +. α + a α

13 ACCURACY AND RELIABILITY OF MODELS OF STOCHASTIC PROCESSES 7 Conclusion Above we obtained some esults fo models of sepaable stationay pocesses belonging to the space Sub ϕ Ω. We found conditions unde which a model appoximates a pocess with a given accuacy and eliability in the Banach space C[, ]. The esults can be extended to the case of othe Banach spaces. Bibliogaphy. Yu. Kozachenko, T. Sottinen, and O. Vasylyk, Simulation of Weakly Self-Simila Stationay Incement Sub ϕ Ω-Pocesses: a Seies Expansion Appoach, Repots of the Depatment of Mathematics, Pepint 398, Univesity of Helsinki, Octobe 4.. Yu. V. Kozachenko and O. A. Pashko, Models of Stochastic Pocesses, Kyiv Univesity, Kyiv, 999. Ukainian 3. Yu. Kozachenko and I. Rozoa, Simulation of stochastic Gaussian pocesses, Random Opeatos Stoch. Equations 3, no. 3, MR987 4i:65 4. V. V. Buldygin and Yu. V. Kozachenko, Metic Chaacteization of Random Vaiables and Random Pocesses, TViMS, Kiev, 998; English tansl. AMS, Povidence,. MR74376 g: Yu. V. Kozachenko and O. I. Vasylyk, On the distibution of supema of Sub ϕ Ω andom pocesses, Theoy Stoch. Pocesses 4 998, no., MR664 4k:694 Depatment of Pobability Theoy and Mathematical Statistics, Faculty fo Mechanics and Mathematics, National Taas Shevchenko Univesity, Academician Glushkov Avenue 6, Kyiv 37, Ukaine addess: yvkuniv.kiev.ua Depatment of Pobability Theoy and Mathematical Statistics, Faculty fo Mechanics and Mathematics, National Taas Shevchenko Univesity, Academician Glushkov Avenue 6, Kyiv 37, Ukaine addess: iozoa@bigmi.net Received 7/FEB/4 Tanslated by OLEG KLESOV

ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS

STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The