Contribution of the Extreme Term in the Sum of Samples with Regularly Varying Tail
aa r X i v : . [ m a t h . P R ] J a n Contribution of the Extreme Term in the Sum ofSamples with Regularly Varying Tail
Van Minh Nguyen ∗ August 13, 2018
Abstract
For a sequence of random variables ( X , X , . . . , X n ), n ≥
1, that areindependent and identically distributed with a regularly varying tail withindex − α , α ≥
0, we show that the contribution of the maximum term M n , max( X , . . . , X n ) in the sum S n , X + · · · + X n , as n → ∞ ,decreases monotonically with α in stochastic ordering sense. Keywords
Extreme · Sum · Regular variation · Stochastic ordering · Wireless network modeling
Mathematics Subject Classification (2000) · · · Let ( X , X , . . . ) be a sequence of random variables that are independent andidentically distributed (i.i.d.) with distribution F . For n ≥
1, the extreme termand the sum are defined as follows: M n , n max i =1 X i , S n , n X n =1 X i . (1)The influence of the extreme term in the sum has various implications in boththeory and applications. In particular, it has been used to characterize thenature of possible convergence of the sums of i.i.d. random variables (Darling1952). On the other hand, besides well-known applications to risk management,insurance and finance (Embrechts et al. et al. F ( x ) , − F ( x ) the tail distribution of F , a primary result ofthis question is due to (Darling 1952) in which it is shown that ∗ Huawei Technologies, France. Email: [email protected] roperty ((Darling 1952)) . Suppose that X i ≥ . Then E ( S n /M n ) → as n → ∞ if for every t > we have lim x →∞ ¯ F ( tx )¯ F ( x ) = 1 . (2)In Darling’s result, condition (2) is of particular interest as it is a specificcase of a more general class called regularly varying tails which is defined in thefollowing (Bingham et al. Definition 1.
A positive, Lebesgue measurable function h on (0 , ∞ ) is called regularly varying with index α ∈ R at ∞ if lim x →∞ h ( tx ) h ( x ) = t α for every constant0 < t < ∞ .In the sequel, R α denotes the class of regularly varying functions with index α , and in particular R is referred to as the class of slowly varying functions. Inaddition, d → , p → , and a.s. → stands for the convergence in distribution , convergencein probability , and almost sure convergence, respectively.Since the work of (Darling 1952), there has been subsequent extensions whichin particular investigated other cases of regularly varying tails. Among those,(Arov & Bobrov 1960) derived the characteristic function and limits of thejointed sum and extreme term for a regularly varying tail. (Teugels 1981) derivedthe limiting characteristic function of the ratio of the sum to order statistics,and moreover investigated norming sequences for its convergence to a constantor a normal law. (Chow & Teugels 1978; Anderson & Turkman 1991, 1995)investigated the asymptotic independence of normed extreme and normed sum.Unlike the slowly varying case, (O’Brien 1980) showed that M n /S n a.s. → ⇔ E X < ∞ . For ¯ F regularly varying with index − α , α >
0, (Bingham & Teugels1981) showed that the extreme term only contributes a proportion to the sum:
Property ((Bingham & Teugels 1981)) . The following are equivalent:1. ¯ F ∈ R − α for < α < ;2. M n /S n d → R where R has a non-degenerate distribution;3. E ( S n /M n ) → (1 − α ) − . Property ((Bingham & Teugels 1981)) . Let µ = E X . The following areequivalent:1. ¯ F ∈ R − α for < α < ;2. ( S n − ( n − µ ) /M n d → D where D has a non-degenerate distribution;3. E (( S n − ( n − µ ) /M n ) → c where c is a constant. (Maller & Resnick 1984) extended Darling’s convergence in mean to conver-gence in probability of S n /M n : 2 roperty ((Maller & Resnick 1984)) . S n /M n p → ⇔ ¯ F ∈ R . The ratio of the extreme to the sum has been further studied with thefollowing result:
Property ((Downey & Wright 2007)) . If either one of the following condi-tions:1. ¯ F ∈ R − α for α > ,2. F has finite second moment,holds, then E ( M n /S n ) = E M n E S n (1 + o (1)) , as n → ∞ . To this end, the contribution of the extreme term in the sum has beeninvestigated and classified in the following cases: • ¯ F ∈ R ; • ¯ F ∈ R − α , 0 < α < • ¯ F ∈ R − α , 1 < α < • ¯ F ∈ R − α , α > consider two cases with ¯ F ∈ R − α and ¯ F ∈ R − α in which ≤ α < α , which case results in larger M n /S n ? This question is partic-ularly important for analysis and design of wireless communication networks.In this context, random variables ( X , X , . . . ) are used to model the signalthat a user receives from base stations. It has been proven that the tail dis-tribution of X i is regularly varying (Nguyen & Kountouris 2017) either dueto the effect of distance-dependent propagation loss or due to fading (Tse &Viswanath 2005) that is regular varying (Rajan et al. M n expresses the useful signal power and( S n − M n ) is the total interference due to the other transmitters (Nguyen 2011). M n / ( S n − M n ) is hence the signal-to-interference ratio (SIR), and its limit as n → ∞ happens for a dense or ultra-dense network (Nguyen & Kountouris 2016;Nguyen 2017). Capacity of a communication channel is expressed in term of thewell-known Shannon’s capacity limit of log(1 + SIR) (Shannon 1948; Cover &Thomas 2006) considering that thermal noise is negligible in comparison to theinterference. Therefore, M n /S n (or S n /M n ) is a fundamental parameter of wire-less network engineering. From the perspective of capacity, a primary purpose3s to design the network such that X i possesses properties that make M n /S n aslarge as possible. In particular, how M n /S n varies according to the tail of X i turns out to be a critical question.In this paper, we establish a stochastic ordering for S n /M n and show thatbetween two cases with ¯ F ∈ R − α and ¯ F ∈ R − α in which 0 ≤ α < α , thecontribution of M n in S n as n → ∞ is larger in the former than in the lattercase in stochastic ordering sense. In the following, for n ≥ R n , S n /M n . (3)We also restrict our consideration to non-negative random variables, i.e., X i ≥ F ∈ R − α with α ≥
0. In the context where α is analyzed, avariable v is written as v α , e.g., write S α,n , M α,n , and R α,n for S n , M n , and R n , respectively. Lemma 1.
For s ∈ C with ℜ ( s ) ≥ , define L R n ( s ) , E (cid:0) e − sR n (cid:1) . If ¯ F ∈ R − α with α ≥ then: L R n ( s ) = e − s φ α ( s ) , n → ∞ , (4) where φ α ( s ) , α Z (1 − e − st ) d tt α . (5) Proof.
Let G ( x , · · · , x n ) be the joint distribution of ( X , · · · , X n ) given that M n = X , it is given as follows: G ( d x , · · · , d x n ) = ( F ( d x ) · · · F ( d x n ) if x = max ni =1 x i . (6)Since ( X , · · · , X n ) are i.i.d., M n = X with probability 1 /n . Thus, the (un-conditional) joint distribution of ( X , · · · , X n ) is nG ( x , · · · , x n ). Hence, L R n ( s ) = E ( e − sR n ) = n Z · · · Z e − s ( x + x + ··· + x n ) /x G ( d x , · · · , d x n ) , and using G from (6), we obtain L R n ( s ) = n Z ∞ Z x · · · Z x e − s n Y i =2 e − sx i /x F ( d x i ) ! F ( d x )= ne − s Z ∞ (cid:18)Z e − st F ( x d t ) (cid:19) n − F ( d x )= ne − s Z ∞ ( ϕ ( x )) n − F ( d x ) , (7)4here ϕ ( x ) , Z e − st F ( x d t ) . (8)Given that ℜ ( s ) ≥
0, we can see that | ϕ ( x ) | ≤ Z | e − st F ( x d t ) | ≤ Z F ( x d t ) = F ( x ) < , for x < ∞ . Hence, Z T ( ϕ ( x )) n − F ( d x ) → , as n → ∞ for T < ∞ . As a result, we only need to consider the contribution of large x in (7) for L R n ( s ). An integration by parts with e − st and F ( x d t ) yields: ϕ ( x ) = 1 − e − s ¯ F ( x ) − Z se − st ¯ F ( xt ) d t = 1 − ¯ F ( x ) + Z se − st (cid:0) ¯ F ( x ) − ¯ F ( tx ) (cid:1) d t. (9)For ¯ F ∈ R − α with α ≥
0, we can write¯ F ( tx ) ∼ t − α ¯ F ( x ) , as x → ∞ , < t < ∞ . Thus Z se − st (cid:0) ¯ F ( x ) − ¯ F ( tx ) (cid:1) d t ∼ ¯ F ( x ) Z se − st (1 − t − α ) d t, as x → ∞ . Using an integration by parts with (1 − t − α ) and d (1 − e − st ) we obtain¯ F ( x ) Z se − st (1 − t − α ) d t = − ¯ F ( x ) Z α (1 − e − st ) d tt α . (10)Put φ α ( s ) , Z α (1 − e − st ) d tt α , and substitute it back in (10) and (9), we obtain ϕ ( x ) = 1 − (1 + φ α ( s )) ¯ F ( x ) , for x → ∞ . Substitute ϕ ( x ) back in the expression of L R n ( s ) in (7), we obtain L R n ( s ) ∼ ne − s Z ∞ (cid:0) − (1 + φ α ( s )) ¯ F ( x ) (cid:1) n − F ( d x ) , as n → ∞ . (11)5ere, we resort to a change of variable with v = n ¯ F ( x ) and obtain: L R n ( s ) ∼ e − s Z n (cid:16) − vn (1 + φ α ( s )) (cid:17) n − d v ( a ) → e − s Z ∞ e − v (1+ φ α ( s )) d v = e − s φ α ( s ) , as n → ∞ where ( a ) is due to the formula (1 + xn ) n → e x as n → ∞ .To present the main result, we use the following notation. For two randomvariables U and V , U is said to be smaller than V in Laplace transform order,denoted by U (cid:22) Lt V , if and only if L U ( s ) = E ( e − sU ) ≥ E ( e − sV ) = L V ( s ) forall positive real number s . Theorem 1.
Let R α ,n and R α ,n be as defined in (3) for ¯ F ∈ R − α with α ≥ and for ¯ F ∈ R − α with α ≥ , respectively. Then α ≤ α ⇒ R α ,n (cid:22) Lt R α ,n , n → ∞ . (12) Proof.
The proof is direct from Lemma 1. By noting that α/t α is increasingwith respect to (w.r.t.) α ≥ t ∈ [0 , φ α ( s ) in (5) is increasingw.r.t. α ≥
0. Thus, L R α,n ( s ) as n → ∞ and ℜ ( s ) > α ≥ ≤ α ≤ α , we have L R α ,n ( s ) ≥ L R α ,n ( s ) as n → ∞ for all s with ℜ ( s ) >
0, thus R α ,n (cid:22) Lt R α ,n as n → ∞ .Theorem 1 dictates that the more slowly ¯ F decays at ∞ , i.e., smaller α , thesmaller is the ratio of the sum to the extreme, thus the bigger is the contributionof the extreme term in the sum. This contribution of the extreme term to thesum increases to the ceiling limit 1 when α gets close to 0 as we have knownfrom (Darling 1952; Maller & Resnick 1984).Since we have established the Laplace transform ordering for R n , an imme-diate application is related to completely monotonic and Bernstein functions.Let us recall: • Completely monotonic functions : A function g : (0 , ∞ ) → R + is said to becompletely monotonic if it possesses derivatives of all orders k ∈ N ∪ { } which satisfy ( − k g ( k ) ( x ) ≥ ∀ x ≥
0, where the derivative of order k = 0 is defined as g ( x ) itself. We denote by CM the class of completelymonotonic functions. • Bernstein functions : A function h : (0 , ∞ ) → R + with d h ( x ) / d x beingcompletely monotonic is called a Bernstein function. We denote by B theclass of Bernstein functions.Note that a completely monotonic function is positive, decreasing and convex,whereas a Bernstein function is positive, increasing and concave.6t is well known that for all completely monotonic functions g , we havethat U (cid:22) Lt V ⇔ E ( g ( U )) ≥ E ( g ( V )), whereas for all Bernstein functions h , U (cid:22) Lt V ⇔ E ( h ( U )) ≤ E ( h ( V )). Hence, we can have a direct corollary ofTheorem 1 as follows. Corollary 1.
With the same notation and assumption of Theorem 1, if ≤ α ≤ α , then: ∀ g ∈ CM : E ( g ( R α ,n )) ≥ E ( g ( R α ,n )) , as n → ∞ , ∀ h ∈ B : E ( h ( R α ,n )) ≤ E ( h ( R α ,n )) , as n → ∞ . Note that h ( x ) = 1, ∀ x >
0, is a Bernstein function, whereas g ( x ) = 1 /x , ∀ x >
0, is a completely monotonic function. For two cases with ¯ F ∈ R − α and¯ F ∈ R − α with 0 ≤ α ≤ α , Corollary 1 gives: E (cid:18) S α ,n M α ,n (cid:19) ≤ E (cid:18) S α ,n M α ,n (cid:19) , n → ∞ , E (cid:18) M α ,n S α ,n (cid:19) ≥ E (cid:18) M α ,n S α ,n (cid:19) , n → ∞ . Application Example
We now can show an application of the results de-veloped above to the context of wireless communication networks. The signal-to-interference ratio SIR as described in the Introduction can be expressed interms of R n as follows: 1SIR = S n − M n M n = R n − Z n . (13)Assume that ¯ F ∈ R − α , α ≥
0, the Laplace transform of Z n can be directlyobtained from that of R n as given by Lemma 1 as follows: L Z α,n ( s ) = e s L R α,n ( s ) = (1 + φ α ( s )) − , n → ∞ , (14)for all s ∈ C , ℜ s >
0. It is easy to see that L Z α,n ( s ) is also decreasing withrespect to α ≥ X i which are ¯ F ∈R − α and ¯ F ∈ R − α with 0 ≤ α ≤ α , we firstly have: Z α ,n (cid:22) Lt Z α ,n , n → ∞ . (15)Then, by noting that functions 1 /x and log(1 + 1 /x ) with x > n → ∞ : E (SIR α ) ≥ E (SIR α ) , (16) E (log(1 + SIR α )) ≥ E (log(1 + SIR α )) . (17)This says that it is beneficial to the communication quality and capacity todesign the network such that the signal received from a transmitting base station X i admits a regularly varying tail with as small variation index as possible (i.e.,as close to a slowly varying tail as possible).7 eferences Anderson, C. W., & Turkman, K. F. 1991. The joint limiting distribution ofsums and maxima of stationary sequences.
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