Directional phantom distribution functions for~stationary random fields
DDirectional phantom distribution functionsfor stationary random fields
Adam Jakubowski ∗ , Igor Rodionov † and Natalia Soja-Kukie(cid:32)la ‡ Nicolaus Copernicus University, Toru´n, Poland Steklov Mathematical Instituteof Russian Academy of Sciences, Moscow, Russian Federation
Abstract
We give necessary and sufficient conditions for the existence of a phantom distributionfunction for a stationary random field on a regular lattice. We also introduce a lessdemanding notion of a directional phantom distribution, with potentially broader area ofapplicability. Such approach leads to sectorial limit properties, a phenomenon well-knownin limit theorems for random fields. An example of a stationary Gaussian random fieldis provided showing that the two notions do not coincide. Criteria for the existence ofthe corresponding notions of the extremal index and the sectorial extremal index are alsogiven.
Keywords: stationary random fields; extreme value limit theory; phantom distribution func-tion; extremal index; Gaussian random fields
MSClassification 2010:
The notion of a phantom distribution function was introduced by O’Brien [19]. Let { X n : n ∈ Z } be a stationary sequence with a marginal distribution function F and partial maxima M n := max { X k : 1 ≤ k ≤ n } , n ∈ N . We say that a distribution function G is a phantomdistribution function for { X n } , ifsup x ∈ R | P ( M n ≤ x ) − G ( x ) n | −−−→ n →∞ . This means that G completely describes asymptotic properties (in law) of partial maxima { X n } . G is also involved in description of asymptotics of higher order statistic of { X n } (see[11] and [21]). ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ m a t h . P R ] A p r f G can be chosen in the form G ( x ) = F θ ( x ), i.e. if for some θ ∈ (0 , x ∈ R (cid:12)(cid:12)(cid:12) P ( M n ≤ x ) − P ( X ≤ x ) θn (cid:12)(cid:12)(cid:12) −−−→ n →∞ , then, following Leadbetter [14], we call θ the extremal index of { X n } . The extremal indexis a popular tool in the stochastic extreme value limit theory (see e.g. [15]). There exist,however, important classes of stationary sequences which admit a continuous phantom dis-tribution function, while the notion of the extremal index is irrelevant in the description ofthe asymptotics of their partial maxima. This holds, for example, when Lindley’s process hassubexponential innovations [1] or when the continuous target distribution of the random walkMetropolis algorithm has heavy tails [20].Existence of a phantom distribution function is a quite common property. Doukhan etal. [4, Theorem 6] show, that any α -mixing sequence with continuous marginals admits acontinuous phantom distribution function. General Theorem 2, ibid. , asserts that a stationarysequence { X n } admits a continuous phantom distribution function if, and only if, there existsa sequence { v n } and γ ∈ (0 ,
1) such thatP( M n ≤ v n ) → γ, (1)and for each T >
Condition B T ( { v n } ) is fulfilled:sup p,q ∈ N ,p + q ≤ T · n (cid:12)(cid:12) P (cid:0) M p + q ≤ v n (cid:1) − P (cid:0) M p ≤ v n (cid:1) P (cid:0) M q ≤ v n (cid:1)(cid:12)(cid:12) → . (2)Notice that Condition B T ( { v n } ) can be satisfied even by non-ergodic sequences (see Theorem4, ibid. ). Condition B T ( { v n } ) was introduced in [9].Another interesting issue is that there are “user-friendly” criteria of existence of a phantomdistribution function for arbitrary (non-stationary) sequences - see [10] and [13, Theorem 3].Such results are particularly useful in investigating Markov chains “starting at the point”. As the previous section shows, the theory of phantom distribution functions for random sequences is essentially closed. It is therefore surprising that the corresponding theory ofphantom distributions for random fields over Z d is still far from being complete.Let Z d be the d -dimensional lattice built on integers with the standard (coordinatewise)partial order ≤ . Let { X n : n ∈ Z d } be a d -dimensional stationary random field with amarginal distribution function F and partial maxima defined for j , n ∈ Z d by the formulae M j , n := max { X k : j ≤ k ≤ n } , if j ≤ n , M j , n := −∞ , if j (cid:54)≤ n . It is also convenient to define M n := M , n , n ∈ Z d . Of course, M n is of interest only if n ∈ N d (here and in the sequel we distinguish between N = { , , . . . } and N = { } ∪ N ).It seems that the first paper that mentions the notion of a phantom distribution functionin the context of random fields is [12]. Following this paper we will say that G is a phantomdistribution function for { X n } , ifsup x ∈ R (cid:12)(cid:12)(cid:12) P ( M n ≤ x ) − G ( x ) n ∗ (cid:12)(cid:12)(cid:12) → , as n → ∞∞∞ (coordinatewise) , (3)2here n ∗ = n · n · . . . · n d , if n = ( n , n , . . . , n d ).Theorem 4.3 ibid. states that m -dependent random fields as well as moving maxima, mov-ing averages and Gaussian fields satisfying Berman’s condition admit a phantom distributionfunction in the above, strong sense. Another family of interesting examples, exploring theidea of a tail field in the context of the extremal index can be found in [23].Note that (3) describes the asymptotic behavior of M n regardless of the way in which n grows to ∞∞∞ = ( ∞ , ∞ , . . . , ∞ ). To make this statement precise, let us define a monotonecurve in N d as a map ψψψ : N → N d such that ψψψ ( n ) → ∞∞∞ , for n = 1 , , . . . ψψψ ( n ) ≤ ψψψ ( n + 1)and ψψψ ( n ) (cid:54) = ψψψ ( n + 1) (hence { ψψψ ( n ) ∗ } is strictly increasing) and, as n → ∞ , ψψψ ( n ) ∗ ψψψ ( n + 1) ∗ → . (4)We will say that G is a phantom distribution function for { X n } along ψψψ , ifsup x ∈ R (cid:12)(cid:12)(cid:12) P ( M ψψψ ( n ) ≤ x ) − G ( x ) ψψψ ( n ) ∗ (cid:12)(cid:12)(cid:12) → , as n → ∞ . (5)Any function G satisfying (5) will be denoted by G ψψψ . Within such terminology we have thefollowing Proposition 1.1.
A stationary random field { X n } admits a continuous phantom distribu-tion function G if, and only if, G is a phantom distribution function for { X n } along every monotone curve. Another consequence of (3) is that if x has the property that G ( x ) n ∗ is a “good” approxi-mation of P ( M n ≤ x ), then it is equally good for all other points m with m ∗ = n ∗ . In otherwords, such x is a function of the class L k = { n ∈ N d ; n ∗ = k } rather, than of n alone. Weformalize this observation by introducing the notion of a strongly monotone field of levels .We will say that v ( · ) : N d → R is strongly monotone, if v m ≤ v n whenever m ∗ ≤ n ∗ . Thisimplies, in particular, that v m = v n , if m ∗ = n ∗ .We are now able to give a multidimensional analog of [4, Theorem 2]. Theorem 1.2.
Let { X n : n ∈ Z d } be a stationary random field. Then { X n } admits acontinuous phantom distribution function if, and only if, the following two conditions aresatisfied. (i) There exist γ ∈ (0 , and a strongly monotone field of levels { v n ; n ∈ N d } such that P ( M n ≤ v n ) → γ, as n → ∞∞∞ . (ii) For every monotone curve ψψψ and every
T > the following Condition B ψψψT ( { v ψψψ ( n ) } ) holds. β ψψψT ( n ) := max p (1)+ p (2) ≤ Tψψψ ( n ) (cid:12)(cid:12)(cid:12) P (cid:0) M p (1)+ p (2) ≤ v ψψψ ( n ) (cid:1) − (cid:89) i ∈{ , } d P (cid:0) M ( p ( i ) ,p ( i ) ,...,p d ( i d )) ≤ v ψψψ ( n ) (cid:1) (cid:12)(cid:12)(cid:12) −−−→ n →∞ . (The quantities p (1) and p (2) under maximum take values in N d ). B ψψψT ( { v ψψψ ( n ) } ) looks complicated but it is based on a simple idea. We shallillustrate it in the two-dimensional case. Notice that for d = 2 we have β ψψψT ( n ) = max p + q ≤ Tψψψ ( n ) (cid:12)(cid:12)(cid:12) P (cid:0) M p + q ≤ v ψψψ ( n ) (cid:1) − P (cid:0) M p ≤ v ψψψ ( n ) (cid:1) P (cid:0) M ( p ,q ) ≤ v ψψψ ( n ) (cid:1) P (cid:0) M ( q ,p ) ≤ v ψψψ ( n ) (cid:1) P (cid:0) M q ≤ v ψψψ ( n ) (cid:1) (cid:12)(cid:12)(cid:12) and, moreover, by the stationarity, P (cid:0) M ( p ,q ) ≤ v ψψψ ( n ) (cid:1) = P (cid:0) M (1 ,p +1) , ( p ,p + q ) ≤ v ψψψ ( n ) (cid:1) ,P (cid:0) M ( q ,p ) ≤ v ψψψ ( n ) (cid:1) = P (cid:0) M ( p +1 , , ( p + q ,p ) ≤ v ψψψ ( n ) (cid:1) ,P (cid:0) M q ≤ v ψψψ ( n ) (cid:1) = P (cid:0) M p + , p + q ≤ v ψψψ ( n ) (cid:1) . It follows that if β ψψψT ( n ) →
0, as n → ∞ , then P (cid:0) M p + q ≤ v ψψψ ( n ) (cid:1) can be approximated by theproduct of the four probabilities for maxima over disjoint blocks, as in Figure 1.Figure 1: Breaking probabilities into blocks as a consequence of Condition B ψψψ T ( { v ψψψ ( n ) } ), for d = 2.By convention, if some coordinate of p or q is 0, then P (cid:0) M p + q ≤ v ψψψ ( n ) (cid:1) breaks intosmaller number of blocks (for d = 2 into 2 or 1 block). Remark . By [12, Theorem 4.3] models exhibiting local dependence (like m -dependent ormax- m -approximable random fields) admit a continuous phantom distribution function andso, by our Theorem 1.2, satisfy Condition B ψψψT ( { v ψψψ ( n ) } ). Remark . Readers familiar with mixing conditions may not like the shape of Condition B ψψψT ( { v ψψψ ( n ) } ) for there is no “separation of blocks”. For example Leadbetter and Rootz´en[16] investigate the asymptotics of maxima of stationary fields under Coordinatewise mixing ,which involves separation of blocks. Ling [17] operates with Condition A1 (also involvingseparation of blocks), which is an adaptation of the well-known Condition D for sequences.Apart from the more complicated form of these conditions (that would be overhelming in d -dimensional considerations), they are essentially not easier in verification. We find the formof Condition B ψψψT ( { v ψψψ ( n ) } ) very useful in theoretical consideration, for it reflects the intuitionof breaking probabilities into product of probabilities over blocks and avoids technicalities.As a good example of how to check Condition B ψψψT ( { v ψψψ ( n ) } ) (in one dimension) may serveTheorems 6-9 in [4]. Remark . Suppose that F is continuous. Choose γ ∈ (0 ,
1) and define the following fieldof levels: v n = inf { x : P ( M n ≤ x ) = γ } . Then { v n } is non-decreasing, we have P ( M n ≤ v n ) → γ , but there is no reason to expectthat it is strongly monotone. 4 .3 Directional and sectorial phantom distribution functions Remark 1.5 signalizes a serious difficulty and suggests that the theory of phantom distributionfunctions (and of the extremal index) in the sense of the strong definition (3) is restricted torandom fields with really short-range dependencies (numerous examples of which are men-tioned in the previous section).It may happen that in some models another, weaker notion is more suitable. This is notan exceptional situation in the theory of random fields. For example, Gut [8] gives strong lawsfor i.i.d. sequences indexed by a sector and Gadidov [7] deals with a similar framework for U -statistics. Motivated by these examples we propose a new notion of a directional phantomdistribution function.Let { ψψψ ( n ) } be a monotone curve. We define the class U ψψψ of monotone curves, being a kindof a “neighbourhood” of ψψψ , as follows. A monotone curve ϕϕϕ belongs to U ψψψ if and only if forsome constant C ≥ n ∈ N ϕϕϕ ( n ) ∈ U ( ψψψ, C ) := (cid:91) j ∈ N d (cid:89) i =1 [ C − ψ i ( j ) , Cψ i ( j )] . An example of U ( ψψψ, C ) is shown in Figure 2.Figure 2: The shaded area is the set U ( ψψψ, C ) ⊂ R for C = 2. Definition . Let { ψψψ ( n ) } be a monotone curve. We will say that a distribution function G is the ψψψ -directional phantom distribution function for { X n } , if G is a phantom distributionfunction for { X n } along every monotone curve belonging to the set U ψψψ . We shall denote the ψψψ -directional phantom distribution function by G ψψψ .Note that we already used the notation G ψψψ to denote the phantom distribution function along ψψψ . But there is no ambiguity. As we shall see in Theorem 1.8 below any phantomdistribution function along ψψψ is automatically the ψψψ -directional phantom distribution functionfor { X n } and conversely. Remark . Let ∆∆∆( n ) = ( n, n, . . . , n ), n ∈ N , denote the diagonal map. Observe that ϕϕϕ belongs to U ∆∆∆ if, and only if, ϕ ( n ) , ϕ ( n ) , . . . , ϕ d ( n ) are of the same order, i.e., 1 /C ≤ ϕ i ( n ) /ϕ j ( n ) < C for some C ≥
1, all i, j ∈ { , , . . . , d } and almost all n ∈ N . It is naturalto call G ∆∆∆ a sectorial phantom distribution function.5 heorem 1.8. Let { X n : n ∈ Z d } be a stationary random field and let ψψψ be a monotonecurve.The following statements (i)-(iii) are equivalent. (i) { X n } admits a continuous phantom distribution function along ψψψ . (ii) { X n } admits a continuous ψψψ -directional phantom distribution function. (iii) There exist γ ∈ (0 , and a non-decreasing sequence of levels { v ψψψ ( n ) } , n ∈ N , such that P ( M ψψψ ( n ) ≤ v ψψψ ( n ) ) → γ, as n → ∞ , (6) and for every T > Condition B ψψψT ( { v ψψψ ( n ) } ) holds.Remark . We have not yet addressed the question that is basic for this section: is thereany model for which there exists a sectorial phantom distribution function while there is noglobal phantom distribution function? The answer is yes, and the example is given in thenext section.
First we shall construct two characteristic functions η ( θ ) and η ( θ ) on R using Polya’s recipe(see [5]). The graph of η over R + is a polygon connecting points: (cid:0) , (cid:1) , (cid:0) , γ (cid:0)
27 ln (cid:0) ln 27 (cid:1) ln 27 −
26 ln (cid:0) ln 28 (cid:1) ln 28 (cid:1)(cid:1) , (cid:0) , γ ln (cid:0) ln 28 (cid:1) ln 28 (cid:1) , (cid:0) , γ ln (cid:0) ln 29 (cid:1) ln 29 (cid:1) , . . . , while the graph of η over R + is defined using a different sequence of points: (cid:0) , (cid:1) , (cid:0) , γ (cid:0) − (cid:1)(cid:1) , (cid:0) , γ (cid:1) , (cid:0) , γ (cid:1) , . . . . The graphs of η and η over R − are obtained by reflection. The positive numbers γ and γ satisfy γ > / , γ (cid:0)
27 ln (cid:0) ln 27 (cid:1) ln 27 −
26 ln (cid:0) ln 28 (cid:1) ln 28 (cid:1) < γ (cid:0) − (cid:1) < − γ γ . (7)The reader may verify that such numbers γ , γ do exist and that the corresponding functions η and η satisfy Polya’s criterion. Therefore both { η ( i ) } i ∈ Z and { η ( j ) } j ∈ Z are positivelydefined. It follows that r ij = η ( i ) η ( j ) (8)is a covariance function on Z . This function satisfies δ := sup ( i,j ) ∈ Z \{ (0 , } r ij < − γ γ < , (9)and for i and j with sufficiently large absolute values we have r ij = γ γ ln ln | i | ln | i | | j | . (10)Let X = { X ( i,j ) , ( i, j ) ∈ Z } be a Gaussian stationary random field with mean zero, unitvariance and covariance function EX ( i,j ) X (0 , = r ij . .4.2 Φ is a sectorial phantom distribution function We shall prove that sup x ∈ R (cid:12)(cid:12)(cid:12) P ( M n ≤ x ) − Φ( x ) n (cid:12)(cid:12)(cid:12) → , as n → ∞ , (11)where n = ( n, n ) = ∆∆∆( n ), M n = max ( i,j ) ∈ [1 ,n ] × [1 ,n ] X ( i,j ) and Φ( x ) is the distribution functionof a standard normal random variable. Applying Theorem 1.8 we will conclude that Φ is a∆∆∆-directional (or sectorial) phantom distribution function for X .As usually, in order to prove (11) it is sufficient to show that for every c > P (cid:0) M n ≤ u n ( c ) (cid:1) = Φ (cid:0) u n ( c ) (cid:1) n + o (1) , (12)where levels { u n ( c ) } are such that n (1 − Φ( u n ( c ))) → c . Note that for n large enoughexp (cid:18) − ( u n ( c )) (cid:19) = √ πcu n ( c ) n (1 + o (1)) ≤ √ πcu n ( c ) n = K ( c ) u n ( c ) n (13)and that u n ( c ) ∼ √ n, as n → ∞ . (14)We have by Berman’s inequality for Gaussian stationary sequences ([15, Corollary 4.2.4]) (cid:12)(cid:12) P (cid:0) M n ≤ u n ( c ) (cid:1) − Φ( u n ( c )) n (cid:12)(cid:12) ≤ L ( δ ) (cid:88) ( i,j ) , ( k,l ) ∈{ , ,...,n } i,j ) (cid:54) =( k,l ) Cov (cid:0) X ( i,j ) , X ( k,l ) (cid:1) exp (cid:0) − ( u n ( c )) r i − k,j − l (cid:1) ≤ L ( δ ) n (cid:88) ≤ i,j ≤ n ( i,j ) (cid:54) =(0 , r ij exp (cid:18) − ( u n ( c )) r i,j (cid:19) , (15)where L ( δ ) is a constant depending only on δ and we have used the stationarity and thefact that r ij > i, j ∈ Z . Repeating the steps of the proof of [15, Lemma 4.3.2], choose α, < α < − δ δ , (see (9)) and split the sum in the last line of (15) in two parts Σ ( n ) = (cid:80) ( i,j ) ∈ A n and Σ ( n ) = (cid:80) ( i,j ) ∈ B n , where A n = {(cid:100) n α (cid:101) , . . . , n } × {(cid:100) n α (cid:101) , . . . , n } and B n = { , , . . . , n } \ ( A n ∪ { } ) . First let us find the asymptotics of the part involving Σ ( n ). We have for large n L ( δ ) n Σ ( n ) ≤ L ( δ ) n (cid:0) n α − (cid:0) (cid:100) n α (cid:101) − (cid:1) exp (cid:18) − ( u n ( c )) δ (cid:19) ≤ L ( δ ) K ( c ) δ n α (cid:18) u n ( c ) n (cid:19) δ by (13) ∼ L ( δ ) K ( c ) δ (4 ln n ) δ n α +3 − δ → , by (14) and the choice of α .Next, let us notice that for i, j ≥ (cid:100) n α (cid:101) and n large enough r ij ≤ ln(ln n α ) (cid:0) ln n α (cid:1) ≤ (cid:0) /α (cid:1) ln ln n (ln n ) . δ (cid:48) n = sup i,j ∈ A n r i,j and using (14) we obtain that δ (cid:48) n ( u n ( c )) →
0, as n → ∞ .Keeping this relation in mind we can proceed as follows.4 L ( δ ) n Σ ( n ) = 4 L ( δ ) n (cid:88) ( i,j ) ∈ A n r ij exp (cid:18) − ( u n ( c )) r ij (cid:19) = 4 L ( δ ) n exp( − ( u n ( c )) ) (cid:88) ( i,j ) ∈ A n r ij exp (cid:18) ( u n ( c )) r ij r ij (cid:19) ≤ L ( δ )( K ( c )) n (cid:16) u n ( c ) n (cid:17) n δ (cid:48) n exp (cid:0) δ (cid:48) n ( u n ( c )) (cid:1) by (13)= 4 L ( δ )( K ( c )) δ (cid:48) n ( u n ( c )) exp( δ (cid:48) n ( u n ( c )) ) → , as n → ∞ . Let us consider the monotone curve ψψψ ( n ) = (cid:0) (cid:98) n/ ln n (cid:99) , (cid:98) ln n (cid:99) (cid:1) , n ∈ N . By Proposition 1.1, it is enough to show that Φ is not a phantom distribution functionfor { X ( i,j ) , ( i, j ) ∈ Z } along ψψψ . Notice that the desired property is in agreement with thestatement of [15, Theorem 6.5.1], for we have ψψψ ( n ) ∗ ∼ n and r ψψψ ( n ) ln n → γ γ > . The structure of random variables M ψψψ ( n ) is however more complicated than just partial max-ima of a stationary Gaussian sequence and therefore we have to perform carefully all compu-tations.We will show first thatsup x ∈ R (cid:12)(cid:12) P (cid:0) M ψψψ ( n ) ≤ x (cid:1) − P (cid:0) (cid:102) M n ≤ x (cid:1)(cid:12)(cid:12) → , as n → ∞ , (16)where for each n ∈ N (cid:102) M n is the maximum of ψψψ ( n ) ∗ standard normal random variables ξ , ξ , . . . , ξ ψψψ ( n ) ∗ with ρ n = cov( ξ i , ξ j ) = γ γ ln n , i (cid:54) = j . As in the case of (11), we have to provethat P (cid:0) M ψ ( n ) ≤ w n ( c ) (cid:1) = P (cid:0) (cid:102) M n ≤ w n ( c ) (cid:1) + o (1) , for sequences of levels { w n ( c ) } such that P (cid:0) (cid:102) M n ≤ w n ( c ) (cid:1) → c ∈ (0 , { w n ( c ) } satisfiesexp (cid:16) − w n ( c ) (cid:17) ≤ K (cid:48) ( c ) w n ( c ) n and w n ( c ) ∼ √ n. (17)By virtue of [15, Theorem 4.2.1], we have (cid:12)(cid:12)(cid:12) P (cid:0) M ψ ( n ) ≤ w n ( c ) (cid:1) − P (cid:0) (cid:102) M n ≤ w n ( c ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ L ( δ ) n (cid:88) ( i,j ) ∈ D n | r ij − ρ n | exp (cid:18) − ( w n ( c )) ω ij (cid:19) , (18)8here D n = { ( i, j ) : 0 ≤ i ≤ n ln n , ≤ j ≤ ln n } \ { (0 , } and ω ij = max { r ij , ρ n } = r ij on D n . Let us split the set of indices D n in three smaller parts, D n = D (1) n (cid:116) D (2) n (cid:116) D (3) n , where D (1) n = { ( i, j ) : 0 ≤ i ≤ n α , ≤ j ≤ ln n } \ { (0 , } , D (2) n = { ( i, j ) : n α < i ≤ n ln n , ≤ j ≤ (ln n ) β } and D (3) n = { ( i, j ) : n α < i ≤ n ln n , (ln n ) β < j ≤ ln n } , where the parameters α and β will be chosen later.By (9) we have δ < (1 − γ ) / (1 + 2 γ ), or, equivalently, 2 γ < (1 − δ ) / (1 + δ ). So we canfind α satisfying 2 γ < α < − δ δ . From (17) we have, as n → ∞ , n (cid:88) ( i,j ) ∈ D (1) n | r ij − ρ n | exp (cid:16) − ( w n ( c )) r ij (cid:17) ≤ n n α ln n exp (cid:18) − ( w n ( c )) δ (cid:19) ≤ ( K (cid:48) ( c )) δ n α +1 ln n (cid:18) w n ( c ) n (cid:19) δ ∼ ( √ K (cid:48) ( c )) δ n α − − δ δ (ln n ) δ δ → . Estimation of the term related to the sum over ( i, j ) ∈ D (2) n is a bit more challenging. Forindices ( i, j ) ∈ D (2) n we have | r ij − ρ n | ≤ r ij ≤ γ α ln ln n ln n =: δ n . Therefore we obtain n (cid:88) ( i,j ) ∈ D (2) n | r ij − ρ n | exp (cid:16) − ( w n ( c )) r ij (cid:17) ≤ γ α n n ln n (ln n ) β ln ln n ln n exp (cid:18) − ( w n ( c )) δ n (cid:19) ≤ γ α ( K (cid:48) ( c )) n (ln n ) β − ln ln n (cid:32) √ nn (cid:33) n δ n = γ α ( K (cid:48) ( c )) (ln n ) β − ln ln n exp (cid:18) γ α ln ln n ln n ln n (cid:19) = γ α ( K (cid:48) ( c )) (ln n ) β +2 γ /α − ln ln n. (19)Because γ < α/