Aequat. Math. 89 (2015), 41–48 c The Author(s) 2014. This article is published with open access at Springerlink.com 0001-9054/15/010041-8 published online July 23, 2014 DOI 10.1007/s00010-014-0279-1
Aequationes Mathematicae
Quasiconvex stochastic processes and a separation theorem Dawid Kotrys and Kazimierz Nikodem Dedicated to Professor J´ anos Acz´el on his 90th birthday
Abstract. Quasiconvex stochastic processes are introduced. A characterization of pairs of stochastic processes that can be separated by a quasiconvex stochastic process and a stability theorem for quasiconvex processes are given. Mathematics Subject Classification (2010). Primary: 26A51 · Secondary: 60G99. Keywords. Quasiconvex stochastic process · Separation by quasiconvex stochastic processes · Stability of quasiconvex stochastic processes.
1. Introduction In [1], Baron et al. proved that two real functions f and g defined on a real interval I can be separated by a convex function if and only if they fulfil the following inequality f λx + (1 − λ)y λg(x) + (1 − λ)g(y), for all x, y ∈ I and λ ∈ [0, 1]. In 1994 Smolarz [11] obtained an analogous result for quasiconvex functions. Namely, he proved that two functions f, g : I → R can be separated by a quasiconvex function if and only if f λx + (1 − λ)y max g(x), g(y) , for all x, y ∈ I and λ ∈ [0, 1] (see also [3]). In this paper we introduce the notion of quasiconvex stochastic processes and present some properties of them. In particular we show that a stochastic process is convex if and only if it is Jensen-convex and quasiconvex. Our main result extends the Smolarz separation theorem to quasiconvex stochastic
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processes. As a consequence we obtain a Hyers-Ulam-type stability result for quasiconvex stochastic processes.
2. Preliminaries Let (Ω, A, P ) be an arbitrary probability space and I ⊂ R be an interval. A function X : Ω → R is called a random variable, if it is A-measurable. A function X : I × Ω → R is called a stochastic process, if for every t ∈ I the function X(t, ·) is a random variable. Recall that a stochastic process X : I × Ω → R is said to be convex, if X λt1 + (1 − λ)t2 , · λX(t1 , ·) + (1 − λ)X(t2 , ·) (a.e.), for all t1 , t2 ∈ I and λ ∈ [0, 1]. A stochastic process X : I × Ω → R is called Jensen-convex, if t + t X(t , ·) + X(t , ·) 1 2 1 2 ,· (a.e.), X 2 2 for all t1 , t2 ∈ I. Convex and Jensen-convex stochastic processes were investigated by many authors and various properties and applications of them can be found in the literature (see, for instance, [5,6,9,10] and the references therein). We say that a stochastic process X : I × Ω → R is quasiconvex, if (a.e.), (1) X λt1 + (1 − λ)t2 , · max X(t1 , ·), X(t2 , ·) for all t1 , t2 ∈ I and λ ∈ [0, 1]. We start our investigation with two simple observations. Observation 1. The following conditions are equivalent: (i) X : I × Ω → R is a quasiconvex stochastic process. (ii) For every random variable A : Ω → R the level set LA = t ∈ I : X(t, ·) A(·) (a.e.) is convex. Proof. Suppose first that X is a quasiconvex stochastic process and fix a random variable A : Ω → R. Let t1 , t2 ∈ LA and λ ∈ [0, 1]. By (1) and the definition of level sets we have X λt1 + (1 − λ)t2 , · max X(t1 , ·), X(t2 , ·) max A(·), A(·) = A(·) (a.e.). Thus λt1 + (1 − λ)t2 ∈ LA , which proves that LA is convex. Assume now that the sets LA are convex for all random variables A. Fix t1 , t2 ∈ I and λ ∈ [0, 1]. Define A(·) = max X(t1 , ·), X(t2 , ·) . Then, of course,
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t1 , t2 ∈ LA , and, by the convexity of LA , we have λt1 + (1 − λ)t2 ∈ LA . It means that the inequality (a.e.) X λt1 + (1 − λ)t2 , · A(·) = max X(t1 , ·), X(t2 , ·) holds and X is quasiconvex.
Observation 2. If a stochastic process X : I × Ω → R is convex, then it is quasiconvex. Proof. By the convexity of X, for all t1 , t2 ∈ I and λ ∈ [0, 1], we have X λt1 + (1 − λ)t2 , · λX(t1 , ·) + (1 − λ)X(t2 , ·) max X(t1 , ·), X(t2 , ·) (a.e.), which shows that the process X is quasiconvex.
Clearly, quasiconvex (as well as Jensen-convex) stochastic processes need not be convex. However, if a stochastic proces is both quasiconvex and Jensenconvex, then it is convex. Proposition 3. Let I be an open interval. A stochastic process X : I × Ω → R is convex if and only if it is quasiconvex and Jensen-convex. Proof. The “only if” part is clear. To prove the “if” part fix t1 , t2 ∈ I, t1 < t2 . By the quasiconvexity of X, for every t ∈ [t1 , t2 ], we have (a.e.). X(t, ·) max X(t1 , ·), X(t2 , ·) This implies that the process X is P -upper bounded on [t1 , t2 ], that is lim
sup {P ({ω ∈ Ω : |X(t, ω)| ≥ n})} = 0.
n→∞ t∈[t ,t ] 1 2
Since X is also Jensen-convex, it follows, by the Bernstein-Doetsch-type theorem, that X is continuous in probability and, consequently, convex (see [6] Theorems 4, 5).
3. Main result At the beginning of this section we would like to recall the definition and basic properties of the essential infimum of a collection of functions. Let (Ω, F, μ) be a measure space and S be a collection of measurable functions f : Ω → R. On R the Borel σ-algebra is used. If S is a countable set, then we may define the pointwise infimum of the functions from S, which will itself be measurable. If S is uncountable, then the pointwise infimum need not be measurable. In this case, the essential infimum can be used. The essential infimum of S, written as ess inf S, if it exists, is a measurable function f : Ω → R satisfying the two following axioms: • f g almost everywhere, for any g ∈ S,
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• if h : Ω → R is measurable and h g almost everywhere for every g ∈ S, then h f almost everywhere. Note that if f is the essential infimum and g : Ω → R is equal to f almost everywhere, then g is also an essential infimum. Conversely, if f and g are both essential infima, then, from the above definition f g and g f , so f = g almost everywhere. It can be shown that for a σ-finite measure μ, the essential infimum of S does exist. Furthermore, there exists a sequence (fn )n∈N in S such that ess inf S = inf fn : n ∈ N . For more details we refer the reader to [2]. The following properties of essential infimum will be useful in the sequel. Lemma 4. Let (Ω, F, μ) be a σ-finite measure space, S be a nonempty collection of measurable real functions defined on Ω, and let g : Ω → R be a measurable function. If ess inf S < g almost everywhere, then there exist sets Ωn ∈ F and functions fn ∈ S for n ∈ N, such that μ Ω\ n∈N Ωn = 0 and fn < g on Ωn , n ∈ N. Proof. By the fact mentioned above there exists a sequence (fn )n∈N of elements of S such that ess inf S = inf fn : n ∈ N . Assume that inf fn : n ∈ N < g on Ω = Ω\Ω0 , where μ Ω0 = 0. By the definition of the infimum, for every ω ∈ Ω there exists n ∈ N such that fn (ω) < g(ω). Define the sets Ωn = ω ∈ Ω : fn (ω) < g(ω) . Then Ωn ∈ F, n∈N Ωn = Ω, and fn < g on Ωn for n ∈ N, which was to be proved. Lemma 5. Let f : Ω → R be a measurable function and S be a family of all measurable functions g : Ω → R such that f < g almost everywhere. Then ess inf S = f . Proof. Clearly ess inf S f almost everywhere. To conclude the proof it is enough to observe that f + n1 ∈ S for all n ∈ N. Now we present our main theorem. It gives a condition under which two stochastic processes can be separated by a quasiconvex stochastic process. Theorem 6. Let X, Y : I × Ω → R be stochastic processes. There exists a quasiconvex stochastic process H : I × Ω → R such that X(t, ·) H(t, ·) Y (t, ·)
(a.e.), t ∈ I,
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if and only if
X λt1 + (1 − λ)t2 , · max Y (t1 , ·), Y (t2 , ·)
(a.e.)
(2)
for all t1 , t2 ∈ I and λ ∈ [0, 1]. Proof. The sufficiency is obvious. To prove the necessity assume that X and Y fulfil (2). Given a random variable A : Ω → R consider the level set LA = t ∈ I : Y (t, ·) A(·) (a.e.) . Let CA = conv LA denote the convex hull of the set LA . Define a stochastic process H : I × Ω → R by H t, · = ess inf A : t ∈ CA . Fix t0 ∈ I and take a random variable A : Ω → R such that t0 ∈ CA . In view of the Caratheodory theorem (cf. [8]) we have t0 = λt1 + (1 − λ)t 2 ,for some t1 , t2 ∈ LA and λ ∈ [0, 1]. Hence Y t1 , · A(·) (a.e.) and Y t2 , · A(·) (a.e.) and, by inequality (2), we obtain X t0 , · = X λt1 + (1 − λ)t2 , · max Y (t1 , ·), Y (t2 , ·) A(·) (a.e.). Since the above inequality holds for any random variable A, such that t0 ∈ CA , we get X t0 , · ess inf A : t0 ∈ CA = H t0 , · (a.e.). Moreover, since for every fixed t0 ∈ I we have t0 ∈ LY (t0 ,·) ⊂ CY (t0 ,·) , we get also H t0 , · = ess inf A : t0 ∈ CA Y t0 , · (a.e.). Now we will show that the stochastic process H is quasiconvex. Fix t1 , t2 ∈ I and λ ∈ [0, 1]. The following cases are possible: (i) H t1 , · H t2 , · (a.e.). (ii) H t1 , · > H t2 , · (a.e.). (iii) The following sets Ω1 and Ω2 have positive measure Ω1 = ω : H t1 , ω H t2 , ω , Ω2 = ω : H t1 , ω > H t2 , ω . Assume first that case (i) holds. Wewill show that there exist sets Ωn,k ∈ A, n, k ∈ N, such that P n,k∈N Ωn,k = 1 and for all n, k ∈ N, H λt1 + (1 − λ)t2 , · H t2 , · almost everywhere on Ωn,k . Take an arbitrary random variable B : Ω → R satisfying (3) H t2 , · = ess inf A : t2 ∈ CA < B(·) (a.e.).
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By Lemma 4 there exist sets Ωn ∈ A and random variables An ∈ A : t2 ∈ CA such that P n∈N Ωn = 1 and for every n ∈ N, An < B on Ωn . Then t2 ∈ CAn = conv t ∈ I : Y t, · An (·) (a.e.) (4) ⊂ conv t ∈ I : Y t, · < B(·) (a.e.) on Ωn . Now, we use the fact that H t1 , · H t2 , · < B · (a.e.) and apply Lemma 4 separately for every set Ω ⊂ Ωn , Ωn,k ∈ A and n , n ∈ N. There exist sets Ωn,k random variables An,k ∈ A : t1 ∈ CA|Ωn such that P k∈N Ωn,k = P Ωn and An,k < B on Ωn,k for n, k ∈ N. Then t1 ∈ CAn,k |Ωn = conv t ∈ I : Y t, · An,k (·) (a.e.) on Ωn ⊂ conv t ∈ I : Y t, · < B(·) (a.e.) on Ωn,k . (5) By (4) and (5), for every n, k ∈ N we obtain t1 , t2 ∈ conv t ∈ I : Y t, · < B(·)
(a.e.)
on
Ωn,k
and consequently λt1 + (1 − λ)t2 ∈ conv t ∈ I : Y t, · < B(·) (a.e.) on Ωn,k . Hence, by the definition of H, we get H λt + (1 − λ)t , · B · (a.e.) on 1 2 Ωn,k . Since the family Ωn,k n,k∈N is countable and P k∈N Ωn,k = P Ωn for n ∈ N and P n∈N Ωn = 1, we also have H λt1 + (1 − λ)t2 , · B · (a.e.) on Ω. Using the fact, that this inequality holds for every random variable B satisfying (3), by Lemma 5 we obtain (a.e.). H λt1 + (1 − λ)t2 , · H t2 , · = max H(t1 , ·), H(t2 , ·) This finishes the the proof in case (i). The proof in case (ii) is analogous, so we omit it. Now assume that case (iii) holds. We consider two processes H1 and H2 being the restrictions of H to I × Ω1 and I × Ω2 , respectively. Then for H1 case (i) occurs and for H2 case (ii) occurs. It means that H1 λt1 + (1 − λ)t2 , · max H1 (t1 , ·), H1 (t2 , ·) (a.e.) on Ω1 and H2 λt1 + (1 − λ)t2 , · max H2 (t1 , ·), H2 (t2 , ·)
(a.e.)
on
Ω2 .
Consequently H λt1 + (1 − λ)t2 , · max H(t1 , ·), H(t2 , ·)
(a.e.)
on
Ω
and the proof is complete.
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4. Hyers–Ulam stability As a direct consequence of Theorem 6 we obtain the following Hyers–Ulamtype stability result for quasiconvex stochastic processes. For the classical Hyers–Ulam theorem see [4]. The stability theorem for quasiconvex functions was obtained in [7] (cf. also [11]). Theorem 7. Let ε be a positive constant. If a stochastic process X : I × Ω → R satisfies the inequality X λt1 + (1 − λ)t2 , · max X(t1 , ·), X(t2 , ·) + ε (a.e.) there exists a quasiconvex stochastic for all t1 , t2 ∈ I and λ ∈ [0, 1], then (a.e.) for every process H : I × Ω → R such that X(t, ·) − H(t, ·) 2ε t ∈ I. Proof. To prove the above theorem it is enough to apply Theorem 6 to the stochastic processes X and X + ε. Hence, there exists a process H1 : I × Ω → R, which is quasiconvex and satisfies X(t, ·) H1 (t, ·) X(t, ·) + ε (a.e.). By taking H(t, ·) = H1 (t, ·) − 2ε we get X(t, ·) − H(t, ·) 2ε (a.e.). This completes the proof. Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
References [1] Baron, K., Matkowski, J., Nikodem, K.: A sandwich with convexity. Math. Pannonica 5/1, 139–144 (1994) [2] Doob, J.L.: Measure theory. Graduate Texts in Mathematics. Springer, Berlin (1993) [3] F¨ org-Rob, W., Nikodem, K., P´ ales, Zs.: Separation by monotonic functions. Math. Pannonica 7/2, 191–196 (1993) [4] Hyers, D.H., Ulam, S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821– 828 (1952) [5] Kotrys, D.: Hermite–Hadamard inequality for convex stochastic processes. Aequat. Math. 83, 143–151 (2012) [6] Nikodem, K.: On convex stochastic processes. Aequat. Math. 20, 184–197 (1980) [7] Nikodem, K.: Approximately quasiconvex functions. C. R. Math. Rep. Acad. Sci. Canada 10, 291–294 (1988) [8] Roberts, A.W., Varberg, D.E.: Convex functions. Academic Press, New York (1973) [9] Shaked, M., Shanthikumar, J.G.: Stochastic convexity and its applications. Adv. Appl.Prob. 20, 427–446 (1988) [10] Skowro´ nski, A.: On some properties of J-convex stochastic processes. Aequat. Math. 44, 249–258 (1992) [11] Smolarz, J.: On some functional inequalities connected with quasiconvex functions. C. R. Math. Rep. Acad. Sci. Canada 16/6, 241–246 (1994)
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Kazimierz Nikodem Department of Mathematics and Computer Science University of Bielsko-Biala Willowa 2, 43-309 Bielsko-Biala Poland e-mail:
[email protected] Dawid Kotrys e-mail:
[email protected] Received: February 6, 2014
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