aa r X i v : . [ m a t h . P R ] A p r A Note on Convergence of Random Variables
Ze-Chun Hu, Ting Ma ∗ and Xiu-Ju Zhu College of Mathematics, Sichuan University
April 7, 2020
Abstract
In this note, convergence of random variables will be revisited. We will give the answersto 5 questions among the 6 open questions introduced in (Convergence rates in the lawof large numbers and new kinds of convergence of random variables,
Communication inStatistics - Theory and Methods , DOI: 10.1080/03610926.2020.1716248), and make somerelated discussions.
Key words:
Strongly uniform convergence, strongly almost sure convergence; strong conver-gence in distribution.
Mathematics Subject Classification (2000)
It is well known that all kinds of convergence of random variables play an important role inprobability and statistics. In this note, we will make some discussions on convergence of randomvariables, and aim to give answers to 5 questions among the 6 open questions introduced in [2].Let (Ω , F , P ) be a probability space and { X, X n , n ≥ } be a sequence of random variables.We have the following kinds of convergence: • { X n , n ≥ } is said to almost surely converge to X , if there exists a set N ∈ F such that P ( N ) = 0 and ∀ ω ∈ Ω \ N, lim n →∞ X n ( ω ) = X ( ω ), which is denoted by X n a.s. −→ X or X n → X a.s. . ∗ Corresponding author: College of Mathematics, Sichuan University, Chengdu 610065, ChinaE-mail address: [email protected] (Z.-C. Hu), [email protected] (T. Ma), [email protected] (X.-J.Zhu) { X n , n ≥ } is said to converge to X in probability, if for any ε >
0, lim n →∞ P ( {| X n − X | ≥ ε } ) = 0, which is denoted by X n P −→ X . • { X n , n ≥ } is said to L p -converge to X ( p >
0) if lim n →∞ E [ | X n − X | p ] = 0, which isdenoted by X n L p −→ X . • { X n , n ≥ } is said to L ∞ -converge to X if lim n →∞ k X n − X k ∞ = 0, which is denoted by X n L ∞ −→ X . • { X n , n ≥ } is said to converge to X in distribution, if for any bounded continuous function f , lim n →∞ E [ f ( X n )] = E [ f ( X )], which is denoted by X n d −→ X . • { X n , n ≥ } is said to completely converge to X , if for any ε > P ∞ n =1 P ( {| X n − X | ≥ ε } ) < ∞ , which is denoted by X n c.c. −→ X (see [1]). • { X n , n ≥ } is said to S- L p converge to X ( p >
0) if P ∞ n =1 E [ | X n − X | p ] < ∞ , which isdenoted by X n S - L p −→ X (see [3, Definition 1.4]). • { X n , n ≥ } is said to strongly almost surely converge to X with order α ( α > P ∞ n =1 | X n − X | α < ∞ a.s., which is denoted by X n S α - a.s. −→ X (see [2, Definition 1.1]). • { X n , n ≥ } is said to strongly L ∞ -converge to X if P ∞ n =1 k X n − X k ∞ < ∞ , which isdenoted by X n S - L ∞ −→ X (see [2, Definition 1.2]). • { X n , n ≥ } is said to S - d converge to X , if for any bounded Lipschitz continuous function f , P ∞ n =1 | E [ f ( X n ) − f ( X )] | < ∞ , which is denoted by X n S - d −→ X (see [2, Definition 1.3]). • { X n , n ≥ } is said to S - d converge to X , if for any continuous point x of F , P ∞ n =1 | F n ( x ) − F ( x ) | < ∞ , which is denoted by X n S - d −→ X , where F n and F are the distribution functionsof X n and X , respectively (see [2, Definition 1.4]).In the final section of [2], the following 6 open questions were introduced: Question 1.
What is the relation between the S - d convergence and the S - d convergence? Question 2.
Does X n S - L ∞ −→ X imply that X n S - d −→ X ? Question 3.
Does X n S - L −→ X imply that X n S - d −→ X ? Question 4.
Does X n S α - a.s. −→ X ( α >
0) imply that X n S - d −→ X ? Question 5.
Does X n c.c. −→ X imply that X n S i - d −→ X for i ∈ { , } ? Question 6.
Can we give a Skorokhod-type theorem for the strong convergence in distributionand the S α - a.s. convergence?In this note, we will give the answers to the first 5 questions, and make some related discussions.For simplicity, we introduce the following definition.2 efinition 1.1 Let { X, X n , n ≥ } be a sequence of random variables. If for any bounded Lips-chitz continuous function f , it holds that P ∞ n =1 E [ | f ( X n ) − f ( X ) | ] < ∞ , then { X n , n ≥ } is saidto S ∗ - d converge to X , which is denoted by X n S ∗ - d −→ X . It is easy to know that X n S - L −→ X ⇒ X n S ∗ - d −→ X ⇒ X n S - d −→ X .Let { X, X n , n ≥ } be a sequence of random variables. Denote by { f ( t ) , f n ( t ) , n ≥ } thecorresponding characteristic functions. It is well known that X n d −→ X if and only if for any realnumber t , f n ( t ) converges to f ( t ) as n → ∞ . In virtue of this result, we introduce the followingdefinition. Definition 1.2
Let { X, X n , n ≥ } be a sequence of random variables. If for any real number t , it holds that P ∞ n =1 (cid:12)(cid:12) E [ e itX n ] − E [ e itX ] (cid:12)(cid:12) < ∞ , then { X n , n ≥ } is said to S - d converge to X ,which is denoted by X n S - d −→ X . It is easy to know that X n S - d −→ X ⇒ X n S - d −→ X . Then we can rewrite the diagram in [2] asfollows: X n S ∗ - d −→ X ⇒ X n S - d −→ X ⇒ X n S - d −→ X ⇑ ⇓ X n S - L ∞ −→ X ⇒ X n S - L −→ X ⇒ X n S - a.s. −→ X X n d −→ X ⇓ ⇓ ⇑ X n c.c. −→ X ⇒ X n a.s. −→ X ⇒ X n P −→ X. ⇑ ⇑ X n L ∞ −→ X ⇒ X n L −→ X The rest of this note is organised as follows. In Section 2, we give the answers to Questions1-5 based on 3 examples and some results in [2]. In Section 3, we make more discussions on therelation among several kinds of convergence of random variables.
As to the answers to the first 5 questions introduced in Section 1, we have
Proposition 2.1 (i) X n S - d −→ X ; X n S - d −→ X , and X n S - d −→ X ; X n S - d −→ X ;(ii) X n S - L ∞ −→ X ; X n S - d −→ X ;(iii) X n S - L −→ X ; X n S - d −→ X ;(iv) X n S α - a.s. −→ X ( α > ) ; X n S - d −→ X ;(v) X n c.c. −→ X ; X n S i - d −→ X for i ∈ { , } . Example 2.2
Let α > . Define Ω = (0 , , F = B (Ω) and P be the Lebesgue measure on Ω .For n ∈ N , we define a random variable X n as follows: X n ( ω ) := (cid:26) , if ω ∈ (0 , n ); n /α , if ω ∈ [ n , . By [2, Example 3.11], we know that X n c.c. −→ , which together with [2, Theorem 3.5] impliesthat X n S - d −→ .In the following, we will show that when α > , X n S - d . Let f ( x ) = sin x . Then f ( x ) is abounded Lipschitz continuous function. We have ∞ X n =1 | E [ f ( X n ) − f (0)] | = ∞ X n =1 | E [sin X n − sin 0] | = ∞ X n =1 | E [sin X n ] | = ∞ X n =1 (cid:20) n sin 1 + (cid:18) − n (cid:19) sin 1 n /α (cid:21) = sin 1 ∞ X n =1 n − ∞ X n =1 n sin 1 n /α + ∞ X n =1 sin 1 n /α . It is easy to know that the first two sums are convergent. By lim n →∞ sin n /α n /α = 1 , and the fact that for α > , P ∞ n =1 1 n /α = ∞ , we know that the sum P ∞ n =1 sin n /α is divergent.Hence ∞ X n =1 | E [ f ( X n ) − f (0)] | = ∞ . It follows that X n S - d . Example 2.3
Define
Ω = (0 , , F = B (Ω) and P be the Lebesgue measure on Ω . Let α, β betwo constants satisfying < α < , β > . Let X be a random variable defined on (Ω , F , P ) withthe density function f ( u ) = (1 − α )(1 − u ) − α , u ∈ (0 , . For any n ∈ N , define a random variable X n := X + 1 n β . Then we have ∞ X n =1 k X n − X k ∞ = ∞ X n =1 n β < ∞ , hich implies that X n S - L ∞ −→ X .Denote by F n and F the distribution functions of X n and X , respectively. Suppose that (1 − α ) β ≤ . Then we have ∞ X n =1 | F n (1) − F (1) | = ∞ X n =1 (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) − n β (cid:19) − F (1) (cid:12)(cid:12)(cid:12)(cid:12) = ∞ X n =1 (cid:20) F (1) − F (cid:18) − n β (cid:19)(cid:21) = ∞ X n =1 Z − nβ (1 − α )(1 − u ) − α du = (1 − α ) ∞ X n =1 Z nβ v − α dv = ∞ X n =1 n (1 − α ) β = ∞ . It follows that X n S - d X . Example 2.4
Define
Ω = (0 , , F = B (Ω) and P be the Lebesgue measure on Ω . For n ∈ N ,we define a random variable X n as follows: X n ( ω ) := (cid:26) , if ω ∈ (0 , n );0 , if ω ∈ [ n , . It is easy to check that for any α > , any ω ∈ (0 , , we have ∞ X n =1 | X n ( ω ) − | α < ∞ . It follows that X n S α - a.s. → .Let f ( x ) = sin x . Then f is a bounded Lipschitz continuous function. We have ∞ X n =1 | E [ f ( X n ) − f (0)] | = ∞ X n =1 | E [sin X n − sin 0] | = ∞ X n =1 | E [sin X n ] | = ∞ X n =1 (cid:12)(cid:12)(cid:12)(cid:12) n sin 1 + (cid:18) − n (cid:19) sin 0 (cid:12)(cid:12)(cid:12)(cid:12) = sin 1 ∞ X n =1 n = ∞ , and thus X n S - d . roof of Proposition 2.1: (i) By Example 2.2, we get that X n S - d −→ X ; X n S - d −→ X . By Example 2.3 and the fact that X n S - L ∞ −→ X ⇒ X n S - L −→ X ⇒ X n S - d −→ X , we obtain that X n S - d −→ X ; X n S - d −→ X .(ii) By Example 2.3, we obtain that X n S - L ∞ −→ X ; X n S - d −→ X .(iii) By Example 2.3 and the fact that X n S - L ∞ −→ X ⇒ X n S - L −→ X , we obtain that X n S - L −→ X ; X n S - d −→ X .(iv) By Example 2.4, we obtain that X n S α - a.s. −→ X ( α > ; X n S - d −→ X .(v) By Example 2.2, we get that X n c.c. −→ X ; X n S - d −→ X . By Example 2.3 and the fact that X n S - L ∞ −→ X ⇒ X n S - L −→ X ⇒ X n c.c. −→ X , we obtain that X n c.c. −→ X ; X n S - d −→ X . In this section, we make more discussions on the relations of several kinds of convergence ofrandom variables.
By Proposition 2.1(ii), we know that generally speaking, X n S - L ∞ −→ X does not imply X n S - d −→ X .But if some addition condition is assumed, we may have that X n S - L ∞ −→ X ⇒ X n S - d −→ X . By [2,Proposition 3.7(i)] and the fact that X n S - L ∞ −→ X ⇒ X n S - L −→ X ⇒ X n c.c. −→ X , we get that if X is a discrete random variable such that { x ∈ R : P ( X = x ) = 0 } is an open subset of R and X n S - L ∞ −→ X , then X n S - d −→ X . The following proposition extends this result. Proposition 3.1
Let { X, X n , n ≥ } be a sequence of random variables and { F, F n , n ≥ } bethe corresponding sequence of distribution functions. If F is locally Lipschitz continuous at eachcontinuous point x of F and X n S - L ∞ −→ X , then X n S - d −→ X. Remark 3.2 (i) By [2, Theorem 3.5], we know that if C is a constant, then X n c.c. −→ C ⇔ X n S - d −→ C. (ii) By [2, Proposition 3.7], we know that if X n c.c. −→ X , and X is a discrete random variablesuch that { x ∈ R : P ( X = x ) = 0 } is an open subset of R , then X n S - d −→ X .(iii) By (ii) and the fact that X n S - L −→ X ⇒ X n c.c. −→ X , we get that if X n S - L −→ X , and X is a discrete random variable such that { x ∈ R : P ( X = x ) = 0 } is an open subset of R , then X n S - d −→ X . roposition 3.3 If X n c.c. −→ X and P ∞ n =1 E [ | X n − X | I {| X n − X | <ε } ] < ∞ for some positive number ε , then X n S ∗ - d −→ X , and thus X n S - d −→ X . Remark 3.4
The condition that P ∞ n =1 E [ | X n − X | I {| X n − X | <ε } ] < ∞ for some positive number ε is necessary in the sense that if X n c.c. −→ X and X is a bounded random variable, then X n S ∗ - d −→ X if and only if P ∞ n =1 E [ | X n − X | I {| X n − X | <ε } ] < ∞ for any positive number ε .Obviously, by virtue of Proposition 3.3, we need only to show the necessity. Suppose that X n c.c. −→ X , X is a bounded random variable, and X n S ∗ - d −→ X . Let M be a positive numbersatisfying | X | ≤ M a.s.. For any positive number ε , define a function as follows: f ε ( x ) := M + ε if x > M + ε ; x if − M − ε ≤ x ≤ M + ε ; − M − ε if x < − M − ε. It is easy to check that f ε is a bounded Lipschitz continuous function. By the definitions of X n S ∗ - d −→ X and f ε , we have ∞ > ∞ X n =1 E [ | f ε ( X n ) − f ε ( X ) | ]= ∞ X n =1 E [ | f ε ( X n ) − f ε ( X ) | I {| X n − X | <ε } ] + ∞ X n =1 E [ | f ε ( X n ) − f ε ( X ) | I {| X n − X |≥ ε } ] ≥ ∞ X n =1 E [ | f ε ( X n ) − f ε ( X ) | I {| X n − X | <ε } ]= ∞ X n =1 E [ | X n − X | I {| X n − X | <ε } ] . In virtue of the S - d convergence, we have the following three questions: Question 7.
What is the relation between the S - d convergence and the S - d convergence? Question 8.
Does X n S α - a.s. −→ X ( α >
0) imply that X n S - d −→ X ? Question 9.
Does X n c.c. −→ X imply that X n S - d −→ X ?Since X n S - d −→ X ; X n S - d −→ X and X n S - d −→ X ⇒ X n S - d −→ X , we get that X n S - d −→ X ; X n S - d −→ X . By checking Example 2.2, we can get that X n S - d −→ X ; X n S - d −→ X .By checking Example 2.4, we get that X n S α - a.s. −→ X ( α > ; X n S - d −→ X .By checking Example 2.2, we get that X n c.c. −→ X ; X n S - d −→ X . Remark 3.5 [2, Example 3.12] shows that if X is nondegenerate, then X n S - d −→ X does not imply X n P −→ X . .2 Proofs Proof of Proposition 3.1.
Suppose that X n S - L ∞ −→ X . Denote α n = k X n − X k ∞ . Then α n ≥ ∞ X n =1 α n < ∞ . (3.1)For any x ∈ R , we have F n ( x ) − F ( x ) = P ( X n ≤ x ) − F ( x )= P ( X + X n − X ≤ x ) − F ( x ) ≤ P ( X ≤ x + α n ) − F ( x )= F ( x + α n ) − F ( x ) , and F n ( x ) − F ( x ) = 1 − P ( X n > x ) − F ( x )= 1 − P ( X + X n − X > x ) − F ( x ) ≥ − P ( X > x − α n ) − F ( x )= F ( x − α n ) − F ( x )= − [ F ( x ) − F ( x − α n )] . It follows that | F n ( x ) − F ( x ) | ≤ ( F ( x + α n ) − F ( x )) + ( F ( x ) − F ( x − α n )) . (3.2)Let x be any continuous point of F . By the assumption, there exist two constants K and δ such that for any u, v ∈ ( x − δ, x + δ ), | F ( u ) − F ( v ) | ≤ K | u − v | . (3.3)Since lim n →∞ α n = 0, there exists N such that for any n > N , we have α n < δ . Then by (3.3),we get that for any n > N , F ( x + α n ) − F ( x ) ≤ Kα n . Then by (3.1), we get that ∞ X n =1 ( F ( x + α n ) − F ( x )) ≤ N X n =1 ( F ( x + α n ) − F ( x )) + K ∞ X n = N +1 α n < ∞ . (3.4)Similarly, we have ∞ X n =1 ( F ( x ) − F ( x − α n )) < ∞ . (3.5)8y (3.2), (3.4) and (3.5), we get that ∞ X n =1 | F n ( x ) − F ( x ) | < ∞ . It follows that X n S - d −→ X. Proof of Proposition 3.3.
Suppose that f is a bounded Lipschitz continuous function. Thenthere exist two constants K, M such that for any x, y ∈ R , we have | f ( x ) | ≤ M and | f ( x ) − f ( y ) | ≤ K | x − y | . Then by the assumptions, we get ∞ X n =1 E [ | f ( X n ) − f ( X ) | ] = ∞ X n =1 E [ | f ( X n ) − f ( X ) | I {| X n − X | <ε } ] + ∞ X n =1 E [ | f ( X n ) − f ( X ) | I {| X n − X |≥ ε } ] ≤ K ∞ X n =1 E [ | X n − X | I {| X n − X | <ε } ] + 2 M ∞ X n =1 P ( | X n − X | ≥ ε ) < ∞ . Thus X n S - d −→ X . Acknowledgments
We are grateful to the support of NNSFC (Grant No. 11771309 andNo. 11871184).
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