Strong law of large numbers for $φ$-sub-Gaussian random variables under sub-linear expectation spaces

Str ong law of large numbers f or φ -sub-Gaussian random variables under sub-linear expectation spaces Nyanga Honda Masasila 1,* and Istv ´ an Fazekas 2 1 Doctoral School of Informatics, Uni versity of Debrecen, Hungary 2 Faculty of Informatics, Uni versity of Debrecen, Hungary Abstract W e introduce the notions of sub-Gaussian and φ -sub-Gaussian random variables in sub-linear ex- pectation spaces. T o av oid the problem caused by the existence of two dif ferent expectations, i.e., the upper expectation and the lo wer expectation, we di vide the definition of the sub-Gaussian prop- erty into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper frame work for extending Zajko wski’ s general result (Zajko wski 2021) to sublinear e xpectation spaces. W ithin our framew ork, we establish a strong law of lar ge numbers for sub-Gaussian sequences. W e present an e xample sho wing the usefulness of our results. K eywords: sub-linear expectations, non-additiv e probabilities, sub-Gaussian random v ariables, Strong laws of lar ge numbers. Intr oduction In the usual probability frame work, the strong la w of lar ge numbers (SLLN) relies on linearity of expectation and additi vity of probability measure. In many modern applications, including risk measures, robust finance, and models with ambiguity , these assumptions may fail. This has moti v ated the dev elopment of sub-linear e xpectation spaces , in which expectations are sub-linear functionals and probabilities (capacities) are sub-additi ve. In this setting, the SLLN takes another form: it asserts that ev ery cluster point of the sequence of empirical av erages lies between the lo wer and upper expectations, with lo wer capacity equal to 1. Formally , v ω ∈ Ω : E ( X 1 ) ≤ lim inf n →∞ 1 n n X k =1 X k ( ω ) ≤ lim sup n →∞ 1 n n X k =1 X k ( ω ) ≤ ˆ E ( X 1 ) ! = 1 , (2.1) where ˆ E denotes the upper (sub-linear) e xpectation, E the lo wer (conjugate) expectation, and v the corresponding lo wer capacity . Building on this frame work, numerous authors ha ve in v estigated versions of the strong law of large numbers (SLLN) under v arious sets of assumptions. In particular , Marinacci 1999 and Maccheroni and Marinacci 2005 established an SLLN for bounded and continuous random v ariables under a totally monotone capacity . Subsequently , Chen, W u, and Li 2013 pro ved an SLLN for independent 1 N.H. Masasila, I. Fazekas random variables assuming finite (1 + α ) th moments of the upper expectation, while Chen 2012 further relaxed the frame work by removing the continuity assumption on the upper capacity . Sub-Gaussian distributions were studied in Kahane 1960. Then it was used to prov e laws of large numbers (see, e.g., Chow 1966, T aylor and Hu 1987 and Buldygin and Kozachenk o 2000); see also Giuliano Antonini, Ngamkham, and V olodin 2013 for related results. Recently , Zajkowski 2021 de veloped a general strong la w for φ -sub-Gaussian sequences under linear expectations. The purpose of the present paper is to extend this frame work to sub-linear expectation spaces. Basic Pr operties of Capacities and Sub-linear Expectations Based on the classical results of Choquet 1954 and subsequent dev elopments by Huber and Strassen 1973, we summarize the properties of capacities. Let Ω be a non-empty set and let F be a σ -algebra of subsets of Ω . Definition 3.1. ˆ V is called a sub-additiv e probability (upper probability or upper capacity), if 1. (Normalized.) ˆ V (Ω) = 1 , ˆ V ( ∅ ) = 0 . 2. (Monotone.) If A, B ∈ F and A ⊂ B , then ˆ V ( A ) ≤ ˆ V ( B ) . 3. (Sub-additiv e.) If A, B ∈ F , then ˆ V ( A ∪ B ) ≤ ˆ V ( A ) + ˆ V ( B ) . 4. (Lower -continuous.) If A n ↑ A , A n ∈ F , then ˆ V ( A n ) ↑ ˆ V ( A ) . Sub-additi vity and lo wer-continuity imply σ -sub-additivity: F or any sequence { A n } ∞ n =1 ⊂ F , ˆ V ∞ [ n =1 A n ! ≤ ∞ X n =1 ˆ V ( A n ) . An ev ent A is called a quasi-sure (q.s.) e vent if ˆ V ( A c ) = 0 , where A c denotes the complement of the e vent A in Ω . The lo wer capacity corresponding to ˆ V can be expressed in terms of ˆ V as v ( A ) = 1 − ˆ V ( A c ) . The set functions ˆ V and v form a pair of conjugate capacities. Example 3.2. The basic method for obtaining a sub-additiv e probability is as follo ws. Let P be a family of usual (that is, σ -additi ve) probabilities on (Ω , F ) . Introduce ˆ V as ˆ V ( A ) = sup Q ∈P Q ( A ) , ∀ A ∈ F . (3.1) It is easy to see that ˆ V satisfies the properties listed in Definition 3.1. The corresponding lower capacity is v ( A ) = inf Q ∈P Q ( A ) , ∀ A ∈ F . 2 N.H. Masasila, I. Fazekas Standard results such as Chebyshe v’ s inequality and the Borel–Cantelli lemma remain valid in this frame work (see, Chen, W u, and Li 2013 and Peng 2019). No w , we list basic properties of the sub-linear expectation ˆ E ; see Peng 2019 for details. W e mention that, in certain papers, ˆ E is called the upper expectation or sub-additi ve expectation. As usual, an extended real-valued function X on Ω is called a random variable if X − 1 ( A ) ∈ F for any Borel set A . W e assume, that there exists a subset H of the random variables and an extended real-v alued function ˆ E [ X ] of X ∈ H so that the assumptions of Definition 3.3 are satisfied. W e shall suppose that the non-negati ve random v ariables belong to H . In Definition 3.3, the case ˆ E [ X ] + ˆ E [ Y ] = ∞ + ( −∞ ) is e xcluded. Definition 3.3. An extended real-valued function ˆ E [ X ] of X ∈ H is called a sub-linear expectation if it satisfies the follo wing properties. 1. Monotone: If X ≤ Y , then ˆ E [ X ] ≤ ˆ E [ Y ] . 2. Constant preserving: ˆ E [ c ] = c , for any c ∈ R . 3. Sub-additiv e: ˆ E [ X + Y ] ≤ ˆ E [ X ] + ˆ E [ Y ] . 4. Positiv e homogeneous: ˆ E [ λX ] = λ ˆ E [ X ] for an y constant λ ≥ 0 . 5. Monotone con ver gence: If X n ↑ X , and X 1 ≥ 0 , then ˆ E [ X n ] ↑ ˆ E [ X ] . (Ω , H , ˆ E ) is called a sub-linear expectation space. If the sub-linear expectation ˆ E [ . ] is giv en in advance, then we can introduce a sub-additiv e prob- ability ˆ V by ˆ V ( A ) = ˆ E [ I A ] , for any e vent A , where I A denotes the indicator of A . Then this ˆ V satisfies the properties gi ven in Definition 3.1. Throughout this paper , we assume that an upper probability ˆ V is gi ven on (Ω , F ) , and a sub-linear expectation is giv en on H so that ˆ V ( A ) = ˆ E [ I A ] . Relations of random variables are considered as quasi-sure relations, e.g., ˆ E [ X ] = ˆ E [ Y ] if X = Y q.s. Example 3.4. The usual method to define a sub-additive expectation is as follows. Let P be a family of probabilities on (Ω , F ) . Let ˆ E [ X ] = sup Q ∈P E Q [ X ] , (3.2) where E Q is the usual expectation corresponding to Q ∈ P such that E Q [ I A ] = Q ( A ) , ∀ A ∈ F . Then, properties listed in Definition 3.3 are satisfied. W e remark that if an abstract sub-linear expectation ˆ E [ . ] is gi ven with the properties listed in Definition 3.3, and certain additional conditions are satisfied, then ˆ E has a representation (3.2). For the precise statement, see Delbaen 2002 and Peng 2010. For X ∈ H , ˆ E ( X ) can be called supermean, whereas E ( X ) = − ˆ E ( − X ) is called submean. By the abov e properties of ˆ E ( X ) , we have E ( X ) ≤ ˆ E ( X ) . If E ( X )  = − ˆ E ( X ) , then X is said to ha ve mean uncertainty . 3 N.H. Masasila, I. Fazekas Sub-Gaussian Random V ariables under Sub-linear Expectations For a usual probability space (Ω , F , P ) , a random variable X : Ω → R is called sub-Gaussian if there exists a constant a ∈ [0 , ∞ ) such that E P  e λX  ≤ exp  a 2 λ 2 2  , for all λ ∈ R . (4.1) Condition (4.1) sho ws that a random variable is sub-Gaussian if and only if its moment generating function is dominated by that of a centered Gaussian random variable with v ariance parameter a 2 . The term sub-Gaussian reflects this Gaussian-type exponential moment control rather than any e xact distributional identity . W e see that condition (4.1) holds only for zero-mean random v ariables. For sub-linear expectation, we shall replace it with tw o one-sided conditions. More generally , we shall do it for φ -sub-Gaussian property . Therefore, we shall introduce the notion of φ -sub-Gaussian random v ariables for sub-linear e xpec- tation spaces. For usual probability spaces, this was studied e.g., in Zajko wski 2021. W e call a real continuous ev en con ve x function φ ( x ) , x ∈ R , an N -function if the following conditions are satisfied: (a) φ (0) = 0 and φ ( x ) is monotone increasing for x > 0 , (b) lim x → 0 φ ( x ) x = 0 and lim x →∞ φ ( x ) x = ∞ . An N -function φ is called a quadratic N -function if, in addition, φ ( x ) = cx 2 for all | x | ≤ x 0 with c > 0 and x 0 > 0 . In our results, we shall use the follo wing quadratic N -function, see Zajkowski 2021. Definition 4.1. F or p ≥ 1 , let φ p ( x ) =        x 2 2 , if | x | ≤ 1 , 1 p | x | p − 1 p + 1 2 , if | x | > 1 . The function φ p can be considered as a standardization of the function | x | p . W e shall see that, for p = 2 , we have the case of the usual sub-Gaussian random v ariables. Definition 4.2. Let φ be a quadratic N -function. A random v ariable ξ is said to be φ -sub-Gaussian under the sub-linear expectation ˆ E if there exist constants a > 0 , m , and m , such that ˆ E e λ ( ξ − m ) ≤ e φ ( aλ ) , for λ > 0 , (4.2) and ˆ E e λ ( ξ − m ) ≤ e φ ( aλ ) , for λ < 0 . (4.3) a > 0 , m , and m are parameters of the φ -sub-Gaussian variable. 4 N.H. Masasila, I. Fazekas For the upper bound, we shall use con ve x conjugate. F or any real function φ ( x ) , x ∈ R , the function φ ∗ ( y ) , y ∈ R , is called the con ve x conjugate of φ , if φ ∗ ( y ) = sup x ∈ R { xy − φ ( x ) } . The following properties are kno wn for the conv ex conjugate, see e.g. Zajko wski 2020. If φ is a quadratic N -function, then φ ∗ is also a quadratic N -function. For p, q > 1 such that 1 p + 1 q = 1 , we have φ ∗ p = φ q . The con ve x conjugation is order -re versing, i.e. if φ 1 ≥ φ 2 , then φ ∗ 1 ≤ φ ∗ 2 . The following scaling property holds. For a > 0 and b  = 0 , let ψ ( x ) = aφ ( bx ) . Then ψ ∗ ( y ) = aφ ∗  y ab  . Lemma 4.3. Assume that (4.2) and (4.3) ar e satisfied. Then for ε > 0 we have ˆ V ( { ξ − m > ε } ∪ { ξ − m < − ε } ) ≤ 2 e − φ ∗ ( ε/a ) . (4.4) Pr oof. Let ε > 0 and λ > 0 . Then, by the Chebyshev inequality and the φ -sub-Gaussian property , we hav e ˆ V ( ξ − m > ε ) = ˆ V  e λ ( ξ − m ) > e λε  ≤ 1 e λε ˆ E  e λ ( ξ − m )  ≤ 1 e λε e φ ( aλ ) = e − ( λε − φ ( aλ )) . (4.5) T o find the optimal upper bound that is of fered by the abov e inequality , we use con ve x conjugate. So we obtain ˆ V ( ξ − m > ε ) ≤ e − φ ∗ ( ε/a ) , for ε > 0 . (4.6) For the other side, let ε < 0 and λ < 0 . Then, by the Chebyshe v inequality and the φ -sub-Gaussian property , we have ˆ V ( ξ − m < ε ) ≤ 1 e λε ˆ E  e λ ( ξ − m )  ≤ e − ( λε − φ ( aλ )) . (4.7) From this inequality , the optimal upper bound is ˆ V ( ξ − m < ε ) ≤ e − φ ∗ ( ε/a ) , for ε < 0 . (4.8) As φ ∗ is an e ven function, we obtain the result. T o obtain the optimal v alue of the parameter a , we need the follo wing quantity . Definition 4.4. For a φ -sub-Gaussian random variable ξ with fixed m and m let τ φ ( ξ ) be defined as τ φ ( ξ ) = inf n a ≥ 0 : ˆ E e λ ( ξ − m ) ≤ e φ ( aλ ) , for λ > 0 , and ˆ E e λ ( ξ − m ) ≤ e φ ( aλ ) , for λ < 0 o . The Main Result The follo wing theorem is an extension of Theorem 2.1 of Zajko wski 2021 to sub-linear expectation. Theorem 5.1. F or some p > 1 , let Z n , n ≥ 1 , be a sequence of φ p -sub-Gaussian random variables with parameter s m and m . Let τ φ ( Z n ) be defined accor ding to Definition 4.4. If ther e e xist positive 5 N.H. Masasila, I. Fazekas numbers c and α such that for every natural number n , the condition τ φ p ( Z n ) ≤ c n − α is satisfied, then ˆ V  n lim inf n →∞ Z n < m o ∪  lim sup n →∞ Z n > m  = 0 (5.1) and v  m ≤ lim inf n →∞ Z n ≤ lim sup n →∞ Z n ≤ m  = 1 . (5.2) Pr oof. Since V ( · ) and v ( · ) are conjugate to each other , (5.2) is equi v alent to (5.1). By Lemma 4.3, for ε > 0 we ha ve ˆ V ( { Z n − m > ε } ∪ { Z n − m < − ε } ) ≤ 2 exp  − φ q  ε τ φ p ( Z n )  , (5.3) where 1 /p + 1 /q = 1 and we applied that φ q = φ ∗ p . By the condition τ φ p ( Z n ) ≤ c n − α , for all suf ficiently large n we ha ve ε τ φ p ( Z n ) > 1 , so using the definition of φ q , we obtain that the right- hand side of inequality (5.3) is majorated by C 0 exp( − K n q α ) , where C 0 = 2 exp  1 q − 1 2  , and K = 1 q  ε c  q . W e no w sho w that the series ∞ X n =1 ˆ V ( { Z n − m > ε } ∪ { Z n − m < − ε } ) is con vergent. It suffices to pro ve the con vergence of ∞ X n =1 exp  − K n β  , where β = q α > 0 . The function f ( x ) = exp( − K x β ) is positiv e and e ventually decreasing. By the integral test, ∞ X n =1 exp( − K n β ) ≤ Z ∞ 0 exp( − K x β ) dx. Performing the substitution t = K x β , we obtain Z ∞ 0 exp( − K x β ) dx = 1 β K − 1 /β Z ∞ 0 t 1 /β − 1 e − t dt = 1 β K − 1 /β Γ  1 β  < ∞ . No w , by the Borel-Cantelli lemma, it follo ws that the ˆ V measure is zero that infinitely man y of the e vents ( { Z n − m > ε } ∪ { Z n − m < − ε } ) occur . As it is true for an y ε > 0 , so (5.1) is true. 6 N.H. Masasila, I. Fazekas Str ong law of large numbers f or independent sub-Gaussian variables Our aim is to obtain a strong la w of lar ge numbers for independent identically distributed sub- Gaussian random variables. Howe ver , our Theorem 6.1 is true for more general situation. In Example 6.2, we construct a plausible model fitting to Theorem 6.1. W e shall use the independence notion giv en in Chen, W u, and Li 2013. The sequence of random v ariables ξ 1 , ξ 2 , . . . is called independent if for each n = 1 , 2 , . . . and each non-neg ativ e measur- able functions f 1 , f 2 , . . . we have ˆ E ( f 1 ( ξ 1 ) f 2 ( ξ 2 ) · · · f n ( ξ n )) = ˆ E ( f 1 ( ξ 1 )) ˆ E ( f 2 ( ξ 2 )) · · · ˆ E ( f n ( ξ n )) . If ξ 1 , ξ 2 , . . . are independent, then they satisfy the follo wing property ˆ E k Y i =1 exp ( λ ( ξ i − m )) ≤ k Y i =1 ˆ E exp ( λ ( ξ i − m )) for any real λ, m, and positiv e integer k . (6.1) No w , let ξ 1 , ξ 2 , . . . be random variables satisfying the sub-Gaussian property (that is they are φ 2 - sub-Gaussian): for fix ed constants σ > 0 , m , and m ˆ E ( e λ ( ξ i − m ) ≤ e ( σ 2 λ 2 / 2) , for λ > 0 , and ˆ E ( e λ ( ξ i − m ) ≤ e ( σ 2 λ 2 / 2) , for λ < 0 . (6.2) No w , let S n = ξ 1 + · · · + ξ n for an y positi ve integer n . W e are ready to pro ve a strong law of large numbers. Theorem 6.1. Let ξ 1 , ξ 2 , . . . be r andom variables satisfying (6.1) and (6.2) with S n defined abo ve. Then ˆ V n lim inf n →∞ S n n < m o ∪ n lim sup n →∞ S n n > m o = 0 . (6.3) Pr oof. W e shall apply Theorem 5.1 with Z n = S n n . Using the the sub-Gaussian property of ξ i and (6.1), ˆ E exp( λ ( Z n − m )) = ˆ E exp λ n n X i =1 ( ξ i − m ) ! = ˆ E n Y i =1 exp  λ n ( ξ i − m )  ≤ n Y i =1 ˆ E exp  λ n ( ξ i − m )  ≤ n Y i =1 exp  λ n  2  σ 2 2  ! = exp  λ 2 n  σ 2 2  (6.4) for λ > 0 . Similarly , ˆ E exp( λ ( Z n − m )) = ˆ E n Y i =1 exp  λ n ( ξ i − m )  ≤ n Y i =1 ˆ E exp  λ n ( ξ i − m )  ≤ n Y i =1 exp  λ n  2  σ 2 2  ! = exp  λ 2 n  σ 2 2  (6.5) for λ < 0 . So in Theorem 5.1, τ φ 2 ( Z n ) ≤ σ n − (1 / 2) and it implies the result. 7 N.H. Masasila, I. Fazekas W e see, that Theorem 6.1 is v alid for independent identically distributed sub-Gaussian random v ariables. In the follo wing example, we shall use the well-kno wn fact that for a random v ariable X having normal distribution with e xpectation m and variance σ 2 > 0 E e λX = e λ 2 σ 2 2 + λm . Example 6.2. Let Ω be the real line and let F be the family of its Borel sets. Let M be an arbitrary non-empty bounded set of real numbers, let m = inf ( M ) and m = sup( M ) . Let σ > 0 be fixed. Let P m denote the normal distribution with expectation m and variance σ 2 . Then (Ω , F , P m ) is a usual probability space for any m ∈ M . Let E m X = R Ω X dP m be the usual e xpectation of the random v ariable X . Let ξ be the identity map: ξ ( ω ) = ω for any ω ∈ Ω (here Ω is the real line). Then ξ has a normal distribution with mean m and variance σ 2 . No w , define the upper e xpectation as ˆ E X = sup { E m X : m ∈ M } and the upper probability as ˆ V ( A ) = sup { P m ( A ) : m ∈ M } . Then ˆ E ξ = m and for the lo wer expectation E ξ = m . For λ > 0 , from equation E m e λξ − m = e λ 2 σ 2 2 + λ ( m − m ) we see that ˆ E e λξ − m = e λ 2 σ 2 2 , so condition (6.2) is satisfied, we have equality there, and m is the optimal constant in that condition. Similarly , for λ < 0 , we can see that ˆ E e λX − m = e λ 2 σ 2 2 so condition (6.2) is satisfied, we hav e equality there, and m is the optimal constant in that condition. If ξ 1 , ξ 2 , . . . are independent random variables, each of them has the same distrib ution as ξ , S n = ξ 1 + · · · + ξ n , then the strong law of lar ge numbers is satisfied, i.e., (6.3) holds. Ho we ver , we prefer the follo wing explicit construction of the sequence ξ 1 , ξ 2 , . . . , for which (6.1) is satisfied. W e shall use the original idea of Huber and Strassen 1973 to construct ne ga- ti vely dependent random variables. Consider copies (Ω ( i ) , F ( i ) , P ( i ) m ) , i = 1 , 2 , . . . of the prob- ability space (Ω , F , P m ) . Then construct their product probability space (Ω ( ∞ ) , F ( ∞ ) , P ( ∞ ) m ) = Q ∞ i =1 (Ω ( i ) , F ( i ) , P ( i ) m ) . Then let ξ i be the i th coordinate random variable, i.e., ξ i ( x 1 , x 2 , . . . ) = x i for any i . Then, under P ( ∞ ) m , the random variables ξ 1 , ξ 2 , . . . are independent and identically dis- tributed, each having distribution P m , that is, a normal distribution with expectation m and variance σ 2 . But we need upper probability and upper expectation. So let ˆ V ( ∞ ) ( A ) = sup m P ( ∞ ) m ( A ) for any e vent A and ˆ E ( ∞ ) ( X ) = sup m E ( ∞ ) m ( X ) for an appropriate random v ariable X . Let f 1 , f 2 , . . . , f n be non-negati ve measurable functions. Then ˆ E ( ∞ ) ( f 1 ( ξ 1 ) f 2 ( ξ 2 )) · · · f n ( ξ n )) = sup m E ( ∞ ) m ( f 1 ( ξ 1 ) f 2 ( ξ 2 )) · · · f n ( ξ n )) = sup m E m ( f 1 ( ξ 1 )) E m ( f 2 ( ξ 2 )) · · · E m ( f n ( ξ n )) ≤ sup m E m ( f 1 ( ξ 1 )) · sup m E m ( f 2 ( ξ 2 )) · · · sup m E m ( f n ( ξ n )) = sup m E ( ∞ ) m ( f 1 ( ξ 1 )) · sup m E ( ∞ ) m ( f 2 ( ξ 2 )) · · · sup m E ( ∞ ) m ( f n ( ξ n )) = ˆ E ( ∞ ) ( f 1 ( ξ 1 )) · ˆ E ( ∞ ) ( f 2 ( ξ 2 )) · · · ˆ E ( ∞ ) ( f n ( ξ n )) . 8 N.H. Masasila, I. Fazekas So (6.1) is satisfied. 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