Exact and asymptotic properties of δ -records in the linear drift model
EExact and asymptotic properties of δ -records in thelinear drift model. R. Gouet , M. Lafuente , F. J. L´opez , and G. Sanz , Departamento de Ingenier´ıa Matem´atica and CMM (UMI 2807, CNRS),Universidad de Chile, Avenida Blanco Encalada 2120,837-0459, Santiago, Chile. Departamento de M´etodos Estad´ısticos, Facultad de Ciencias, Universidad deZaragoza, C/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain. Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI), Universidad deZaragoza, 50018 Zaragoza, Spain.E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract.
The study of records in the Linear Drift Model (LDM) has attractedmuch attention recently due to applications in several fields. In the present paperwe study δ -records in the LDM, defined as observations which are greater than allprevious observations, plus a fixed real quantity δ . We give analytical properties ofthe probability of δ -records and study the correlation between δ -record events. Wealso analyse the asymptotic behaviour of the number of δ -records among the first n observations and give conditions for convergence to the Gaussian distribution. As aconsequence of our results, we solve a conjecture posed in J. Stat. Mech. 2010, P10013,regarding the total number of records in a LDM with negative drift. Examples ofapplication to particular distributions, such as Gumbel or Pareto are also provided.We illustrate our results with a real data set of summer temperatures in Spain, wherethe LDM is consistent with the global-warming phenomenon. Keywords : Exact results, Extreme value, Stochastic processes
Contents1 Introduction. 22 δ -records in the linear drift model 33 Properties of the δ -record probabilities 4 p δ ( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Continuity of p δ ( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 a r X i v : . [ m a t h . S T ] J un xact and asymptotic properties of δ -records in the linear drift model. N n,δ δ -records . . . . . . . . . . . . . . . . . 136.2 Growth of N n,δ to infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 p δ ( c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.2 Finiteness of the number of δ -records . . . . . . . . . . . . . . . . . . . . 2110.3 Proof of law of large numbers for N n,δ ( c ) . . . . . . . . . . . . . . . . . . 2310.4 Proof of central limit theorem for N n,δ ( c ) . . . . . . . . . . . . . . . . . 2310.5 Correlations in the Pareto Distribution . . . . . . . . . . . . . . . . . . . 28
11 References 291. Introduction.
Extreme values and records have attracted large efforts and attention since thebeginnings of statistics and probability, due to their intrinsic interest and theirmathematical challenges. An important motivation for studying records comes fromtheir connections with other interesting problems and, of course, from their countlesspractical applications in different fields such as climatology [1–4], sports [5–7], finance[8,9] or biology [10]. Moreover, records have been used in statistical inference because, insome contexts, data is inherently composed of record observations [11–14]. The classicalprobabilistic setting of independent and identically distributed random observations (iid)observations has been profusely studied. Main results in this framework can be foundin the monographs [15–17]. In the last years there has been an increasing interest in thestudy of records in correlated observations such as random walks or time series [18–23].An interesting departure from the iid model, which introduces time-dependencebetween observations, results from adding a deterministic linear trend to the iidobservations, thus obtaining the so named Linear Drift Model (LDM). This model wasfirst introduced in [24] and later developed in [25–27]. The model was also considered xact and asymptotic properties of δ -records in the linear drift model. δ -exceedance records [35, 36] have been proposed recently. We will work with δ -records, first introduced in [37], which are observations greater than all previous entries,plus a fixed quantity δ . In the iid setting, the distribution [38, 39], process structure [40]and asymptotic properties [41] of δ -records have been studied. In the case δ <
0, where δ -records are more numerous than records, their use in statistical inference has beenrecently proposed and positively assessed; see [41–43].In this work, we study δ -records from observations obeying the LDM, whilerevisiting some open questions about records. We analyse the positivity and continuityof the asymptotic δ -record probability as a function of δ and of the trend parameter c . We also obtain a law of large numbers and a central limit theorem for the countingprocess of δ -records, thus extending the corresponding results in [24]. Furthermore, wecompletely characterize the finiteness of the number of δ -records and, in particular, wesolve a conjecture posed in [28], about the finiteness of the number of usual records inthe LDM with negative trend.We assess the effect of δ on the δ -record probabilities and correlations, for explicitlysolvable models. Some of the results obtained in these examples are new and shed lighton the behaviour of record events, when the underlying distribution is heavy-tailed.Finally we illustrate our results by analyzing a real dataset of temperatures, which fitsthe LDM with a trend parameter consistent with the global-warming phenomenon. δ -records in the linear drift model Our objects of interest in this paper are δ -records, formally defined as follows: given asequence of observations ( Y n ) n ≥ and δ ∈ R a parameter, Y is defined conventionallyas δ -record and, for j ≥ Y j is a δ -record if Y j > max { Y , . . . , Y j − } + δ .Note that δ -records are just (upper) records, if δ = 0. If δ >
0, a δ -record isnecessarily a record and δ -records are a subsequence of records. On the other hand, if δ <
0, a δ -record can be smaller than the current maximum, so records are a subsequenceof δ -records.Throughout this paper we assume that the Y n are random variables obeying theLDM, that is, Y n can be represented as Y n = X n + cn, n ≥ , (1)where c ∈ R is the trend parameter and ( X n ) n ≥ is a sequence of iid random variables,with (absolutely continuous) cumulative distribution function (cdf) F and probability xact and asymptotic properties of δ -records in the linear drift model. f . Another important parameter of the model is the right-tailexpectation of the X j , defined as µ + = (cid:90) ∞ xf ( x ) dx. For simplicity, we assume the existence of an interval of real numbers I = ( x − , x + ), with −∞ ≤ x − < x + ≤ ∞ , such that f ( x ) >
0, for all x ∈ I , and f ( x ) = 0 otherwise. Notethat x − = inf { x : F ( x ) > } and x + = sup { x : F ( x ) < } .Let 1 j,δ denote the indicator of the event { Y j is a δ -record } . That is, 1 j,δ = 1 if Y j > max { Y , . . . , Y j − } + δ and 1 j,δ = 0 otherwise. So, the number of δ -records up toindex n is computed as N n,δ = (cid:80) nj =1 j,δ .Under the LDM, the probability of { Y j is a δ -record } is easily computed byconditioning, as p j,δ := E [1 j,δ ] = (cid:90) ∞−∞ j − (cid:89) i =1 F ( x + ci − δ ) f ( x ) dx, where E [ · ] denotes the mathematical expectation. Moreover, the asymptotic δ -recordprobability is given by the formula p δ := lim n →∞ p n,δ = (cid:90) ∞−∞ ∞ (cid:89) i =1 F ( x + ci − δ ) f ( x ) dx, (2)which is mathematically justified by the monotone convergence theorem for integrals.In what follows we occasionally write 1 j,δ ( c ) , N n,δ ( c ) , p j,δ ( c ) , p δ ( c ), etc. to emphasizethe dependence on the trend parameter c .
3. Properties of the δ -record probabilities We begin with a simple property about the asymptotic δ -record probability of anaffine transformation of the LDM. Let ˜ X n = bX n + a , with b > a ∈ R , and˜ Y n = ˜ X n + cn, n ≥
1. If ˜ p δ ( c ) is the δ -record probability in this model, then it holds˜ p δ ( c ) = p δb ( cb ) . We consider next some analytical properties of p j,δ ( c ) and p δ ( c ), as functions of c and δ . We note first that both are increasing in c and decreasing in δ . Moreover, it iseasy to see that p j,δ ( c ) is decreasing in j and continuous in c , converging to 1 as c → ∞ .The continuity of p δ ( c ) is less clear because of the infinite product within the integralin (2). p δ ( c )We show that the positivity of p δ ( c ) depends on c and δ and on the right-tail behaviourof F . We consider two cases depending on µ + : xact and asymptotic properties of δ -records in the linear drift model. . µ + = ∞ . In this case p δ ( c ) = 0, for all δ, c ∈ R .To justify this claim, we show that (cid:81) ∞ j =1 F ( x + cj − δ ) = 0 , for all x ∈ ( x − , x + ).If c < F ( x + cj − δ ) →
0, as j → ∞ .If c = 0, we note that µ + = ∞ implies x + = ∞ and so, F ( x − δ ) <
1. Thus (cid:81) ∞ j =1 F ( x + cj − δ ) = 0.Finally, if c >
0, we note that µ + = ∞ implies (cid:80) ∞ i =1 (1 − F ( x + ci − δ )) = ∞ , whichin turn implies (cid:81) ∞ j =1 F ( x + cj − δ ) = 0. This follows from the definition of µ + and fromTaylor’s expansion of log(1 + x ).Distributions with µ + = ∞ can be considered as “right-heavy-tailed” and weobserve that, for such distributions, the linear trend has no impact on the asymptoticprobability of a δ -record. This class of distributions includes the Pareto and Fr´echet,with shape parameter α ∈ (0 , . µ + < ∞ . As in the previous case, we have three situations depending on the sign of c . For c < p δ ( c ) = 0, for all δ ∈ R , since (cid:81) ∞ j =1 F ( x + cj − δ ) = 0, for all x ∈ ( x − , x + ).If c = 0, p δ (0) = (cid:90) ∞−∞ ∞ (cid:89) j =1 F ( x − δ ) f ( x ) dx = (cid:90) ∞ x + + δ f ( x ) dx, (3)which is positive if and only if x + < ∞ and δ < c >
0, then p δ ( c ) = 0 if and only if x + − x − ≤ δ − c . Indeed, notethat, if x + − x − ≤ δ − c , then P [ Y n > Y n − + δ ] = 0, for all n , and so, only the firstobservation (by convention) is a δ -record. Conversely, if x + − x − > δ − c , then theinterval J := ( x − , x + ) ∩ ( x − − c + δ, ∞ ) is nonempty and, for every x ∈ J , we have F ( x + cj − δ ) ≥ F ( x + c − δ ) >
0, for all j . Now, since F ( x + cj − δ ) → j → ∞ , and µ + < ∞ , we have (cid:80) ∞ j =1 (1 − F ( x + cj − δ )) < ∞ , which implies (cid:81) ∞ j =1 F ( x + cj − δ ) > p δ ( c ) > Theorem 1 p δ ( c ) > if and only if µ + < ∞ and one of the following conditions holds(i) c > and δ < x + − x − + c ,(ii) c = 0 , δ < and x + < ∞ .3.2. Continuity of p δ ( c )As commented at the beginning of this section, the continuity of p δ ( c ) is not obvious.However, thanks to Theorem 1 we can restrict attention to distributions F with finiteright-tail expectation since, otherwise, p δ ( c ) vanishes and continuity is trivial. Thus, weassume throughout this section that µ + < ∞ .A first interesting fact, which is rigorously proved in Proposition 6 of the Appendix,is that (cid:81) ∞ i =1 F ( x + ci − δ ) is continuous at every c (cid:54) = 0, for every x ∈ ( x − , x + ), such that x (cid:54) = x − + δ − c . Then, thanks to the bounded convergence theorem of integration, weconclude that p δ ( c ) is continuous, at every c (cid:54) = 0. xact and asymptotic properties of δ -records in the linear drift model. c = 0 is subtler to establish and depends of the sign of δ and thefiniteness of x + , the right-end point of F . Note that, for every c > N ≥
1, wehave ∞ (cid:89) j =1 F ( x − δ ) ≤ ∞ (cid:89) j =1 F ( x + cj − δ ) ≤ N (cid:89) j =1 F ( x + cj − δ ) . Then, taking the limit as c → + in the above inequalities, ∞ (cid:89) j =1 F ( x − δ ) ≤ lim c → + ∞ (cid:89) j =1 F ( x + cj − δ ) ≤ F ( x − δ ) N . Therefore, lim c → + (cid:81) ∞ j =1 F ( x + cj − δ ) is 0, if x < x + + δ , and 1 otherwise. Then, bythe dominated convergence theorem,lim c → + p δ ( c ) = (cid:90) ∞−∞ lim c → + ∞ (cid:89) j =1 F ( x + cj − δ ) f ( x ) dx = (cid:90) ∞ x + + δ f ( x ) dx. Thus, p δ ( c ) is right-continuous at c = 0 by (3). Regarding left-continuity at 0, recallthat p δ ( c ) = 0 for c <
0. So, p δ ( c ) is discontinuous at 0 if and only if x + < ∞ and δ < p δ ( c ) as a function of δ . The result is trivial if c < p δ ( c ) = 0, for all δ ∈ R . For c = 0, note that, by (3), p δ (0) = 1 − F ( x + + δ ), whichis continuous since F is a continuous function.If c > δ n ) n ≥ is a sequence converging to δ , we prove thatlim n →∞ ∞ (cid:89) i =1 F ( x + ci − δ n ) = ∞ (cid:89) i =1 F ( x + ci − δ ) , (4)for all x ∈ ( x − , x + ), x (cid:54) = x − + δ − c . Indeed, let x < x − + δ − c , then F ( x + c − δ ) = 0yielding (cid:81) ∞ i =1 F ( x + ci − δ ) = 0. Also F ( x + c − δ n ) = 0 for n large enough and (4)follows. Let now x > x − + δ − c and ε > x + c − δ − ε > x − . Then, for n large enough, we have | δ n − δ | < ε and − ∞ (cid:88) i =1 log F ( x + ci − δ n ) ≤ − ∞ (cid:88) i =1 log F ( x + ci − ( δ + ε )) < ∞ , since µ + < ∞ . So (4) holds, and continuity follows.In the following theorem we summarize conditions for continuity of p δ ( c ). Theorem 2
The asymptotic δ -record probability p δ ( c ) , as a function of c, δ , is(a) continuous at every c (cid:54) = 0 and right-continuous at c = 0 , for all δ ;(b) discontinuous at c = 0 if and only if x + < ∞ , δ < , and(c) continuous in δ , for all c .xact and asymptotic properties of δ -records in the linear drift model. - - δ p δ ( c ) c = = = c p δ ( c ) δ =- δ =- δ = δ = δ = Figure 1:
Asymptotic δ -record probability p δ ( c ) for the Gumbel distributionas function of δ and c .
4. Exactly solvable models
In general it is not possible to compute exactly the probabilities p j,δ or p δ . We showbelow explicit results for the Gumbel distribution and for particular instances of theDagum family of distributions. Let F ( x ) = exp( − exp( − x )), for x ∈ R , be the Gumbel distribution. Note that F ( x + cj − δ ) = F ( x ) e − cj + δ . Then, if c (cid:54) = 0, n − (cid:89) j =1 F ( x + cj − δ ) = F ( x ) (cid:80) n − j =1 e − cj + δ = F ( x ) e δ e − c − e − nc − e − c and, if c = 0, (cid:81) n − j =1 F ( x + cj − δ ) = F ( x ) ( n − e δ . So, from (2) we get p n,δ ( c ) = (cid:90) ∞−∞ F ( x ) e δ e − c − e − nc − e − c f ( x ) dx = 1 − e − c − e − c + e δ ( e − c − e − nc ) , if c (cid:54) = 0, and p n,δ (0) = 1( n − e δ + 1 . Note that, taking limits as n → ∞ , in the above formulas, we obtain p δ ( c ) = 1 − e − c e δ e − c + 1 − e − c = 11 + e − c − e − c e δ , if c > p δ ( c ) = 0, if c ≤
0, as expected from Theorem 1.Also, for every c > p δ ( c ) decreases with δ as a logistic function of − δ . Figure 1shows the behaviour of p δ ( c ) as a function of δ and c . xact and asymptotic properties of δ -records in the linear drift model. The random variables in the Dagum family of distributions have cdf given by F ( x ) = (cid:16) (cid:0) xb (cid:1) − a (cid:17) − q , for x ≥
0, and F ( x ) = 0, for x <
0, where a, b, q are positive parameters.Two important distributions in the Dagum family are the Loglogistic, with parameters a, b > q = 1, and the Pareto (up to a shift), with a > b = q = 1. For simplicity, inthis example we limit our attention to the case a = 1, which has µ + = ∞ .By Theorem 1 we know that p δ ( c ) = 0, for every c, δ ∈ R , so we chose to analysethe speed of convergence of p n,δ ( c ) to 0, for some values of c, δ . To that end, observethat the formula for p n,δ takes the manageable form p n,δ ( c ) = (cid:90) ∞ ( δ − c ) + n − (cid:89) i =1 (cid:18) x + ci − δx + b + ci − δ (cid:19) q f ( x ) dx, (5)which becomes simpler if we further assume that c = b (that is, the trend parameter ofthe LDM is equal to the scale parameter of the distribution). From (5) we get p n,δ ( c ) = (cid:90) ∞ ( δ − c ) + (cid:18) x + c − δx + cn − δ (cid:19) q f ( x ) dx. (6)Note that the Pareto(1,1) distribution, taking c = 1, is included as a particular case.This distribution will be studied at the end of this example and later, in section 5, inthe context of δ -record correlations.We introduce the notation p ( q ) n,δ ( c ) to make explicit the dependence of p n,δ ( c ) on q .First, for records ( δ = 0) we have, p ( q ) n, ( c ) = cq (cid:90) ∞ x q − ( x + cn ) − q ( x + c ) − dx = qn − q (cid:90) t q − (1 − t ( n − /n ) − q dt (7)= q ( n − q (cid:90) n ( y − q − y dy, (8)where the second equality follows from the change of variable x = ct/ (1 − t ) and thethird from 1 − t ( n − /n = 1 /y .Observe that (7) and (8) do not depend on c and so, for the sake of simplicity, wewrite p ( q ) n, . Moreover, from formula (7) we see that p ( q ) n, = n − q F ( q, q ; q + 1; ( n − /n ) , where F is the Gauss hypergeometric function.Also, from (8) and using the binomial expansion, for q = 1 , , . . . , we readily obtain p ( q ) n, = q ( n − q (cid:32) ( − q − log n + q − (cid:88) k =1 (cid:18) q − k (cid:19) ( − q − − k k ( n k − (cid:33) . (9) xact and asymptotic properties of δ -records in the linear drift model. p ( q ) n, , for any q ∈ (0 , ∞ ), can be obtained from (8).For q = 1, (9) yields p (1) n, = n − log n . For q >
1, the leading term in the integral in (8)is y q − , so p ( q ) n, ∼ qq − n . For q ∈ (0 , p ( q ) n, ∼ n − q q (cid:90) ∞ ( y − q − y dy = n − q q Γ(1 − q )Γ( q ) . Thus, p ( q ) n, ∼ n − q q Γ(1 − q )Γ( q ) , if 0 < q < , log( n ) /n, if q = 1 ,n − qq − , if q > . (10)It is interesting to observe that the limiting behaviour of p ( q ) n, , as a function of thepower of the tail q , seems to match the asymptotic behaviour of p n, ( c ) when F is theFr´echet distribution ( F ( x ) = exp( − x − ), x >
0) and the tuning parameter is the trend c , studied in [27].We now consider δ (cid:54) = 0 and investigate whether p ( q ) n,δ /p ( q ) n, →
1, as n → ∞ . Thisresult can be expected since, as µ + = ∞ , the variables X n take very large values, so δ may have little influence on the probability of δ -record, in the long term.From (6) we may evaluate p ( q ) n,δ , for any q ∈ N , although the computation becomeslengthy as q grows. We have carried out the computation with values of q from 1 to 7,and obtained p (1) n,δ ∼ log( n ) n , p ( q ) n,δ ∼ qq − n , q = 2 , . . . , . So, from (10) we have p ( q ) n,δ /p ( q ) n, →
1, at least for q = 1 , . . . , q ∈ (0 , ∞ ), the limit behaviour of (6) is harder to analyse.To get a tractable expression, we impose δ = c . Proceeding as above, we have, for n > p ( q ) n,δ = q ( n − q ( n − q (cid:90) n − ( y − q − y q +1 dy. Therefore, we have p ( q ) n,δ ∼ n − q Γ(2 q )Γ(1 − q )Γ( q ) , if 0 < q < , log( n ) /n, if q = 1 ,n − qq − , if q > . So, under the above stated conditions, p ( q ) n,δ ∼ p n, , for q ≥
1, but this is not the case if q ∈ (0 , F ( x ) = (1 − /x ) 1 { x> } , and take c = 1. The probability of δ -record is explicitly xact and asymptotic properties of δ -records in the linear drift model. - - δ p n , δ ( ) n = = = = n p n , δ ( ) δ =- δ =- δ = δ = δ = Figure 2: δ -record probability p n,δ ( c ) for the Pareto Distribution as afunction of δ and n with c = 1. computed as: p n,δ = (cid:90) ∞ δ ∨ x − δx ( x + n − − δ ) dx = 1( n − − δ ) (cid:16) ( n −
1) log( n − min { ,δ } max { ,δ } ) − min { , δ } ( n − − δ ) (cid:17) , (11)if δ (cid:54) = n − p n,δ = n − , if δ = n −
1. Figure 2 shows the behaviour of p n,δ as afunction of n and δ .
5. Correlations
The indicators of δ -records are in general not independent in the case of iid randomvariables, see [41]. In [31] the authors study the dependence of record events in theLDM, by means of the following dependence index ( δ = 0 in their case) l n ( c, δ ) := P [obs. n and n + 1 are δ -records] P [obs. n is δ -record] P [obs. n + 1 is δ -record] = E [1 n,δ n +1 ,δ ] E [1 n,δ ] E [1 n +1 ,δ ] . If the events are independent, then l n ( c, δ ) = 1. Otherwise, values greater or smallerthan 1 indicate positive or negative correlation, respectively. That is, neighbouring δ -records tend to attract or repel each other, if l n > l n < E [1 n,δ n +1 ,δ ] we consider the decomposition E [1 n,δ n +1 ,δ ] = E [1 n,δ n +1 ,δ { Y n
0, can be written as E [1 n,δ n +1 ,δ ] = ∞ (cid:90) −∞ (cid:32) ∞ (cid:90) s − c n − (cid:89) j =1 F ( s + cj − δ ) f ( t ) dt + s − c (cid:90) s − c + δ n (cid:89) j =2 F ( t + cj − δ ) f ( t ) dt (cid:33) f ( s ) ds = ∞ (cid:90) −∞ (cid:32) (1 − F ( s − c )) n − (cid:89) j =1 F ( s + cj − δ ) + s − c (cid:90) s − c + δ n (cid:89) j =2 F ( t + cj − δ ) f ( t ) dt (cid:33) f ( s ) ds, (13) xact and asymptotic properties of δ -records in the linear drift model. δ ≥ E [1 n,δ n +1 ,δ ] = ∞ (cid:90) −∞ ∞ (cid:90) s − c + δ n − (cid:89) j =1 F ( s + cj − δ ) f ( t ) dtf ( s ) ds = ∞ (cid:90) −∞ (1 − F ( s − c + δ )) n − (cid:89) j =1 F ( s + cj − δ ) f ( s ) ds, (14)since the second term in (12) vanishes.As for E [1 n,δ ], it is not possible to explicitly compute E [1 n,δ n +1 ,δ ], in general.Nevertheless, it is still possible to describe the behaviour of the dependence index insome particular cases. Let c > F the Gumbel distribution, as in section 4.1. When δ < n → ∞ ,elementary but lengthy computations yieldlim n →∞ E [1 n,δ n +1 ,δ ] = ( e c − ( e c − e δ + 1)( e c + e δ − e c + e δ − l ∞ ( c, δ ) := lim n →∞ l n ( c, δ ) = ( e c + e δ − e c − e δ + 1)( e c + e δ − . By differentiating with respect to c , we see that l ∞ ( c, δ ) is decreasing in c and boundedbelow by 1, since lim c →∞ l ∞ ( c, δ ) = 1. With respect to δ we find that the derivative ∂l ∞ ∂δ vanishes at δ = log(1 − e c + √ e c − e c ) , and then, for any c ,max δ< l ∞ ( c, δ ) = 2 e c (cid:0)(cid:112) e c ( e c − − e c + 1 (cid:1)(cid:112) e c ( e c −
1) = 2 (cid:0) e c − (cid:112) e c sinh ( c ) (cid:1) . Note also that lim δ →−∞ l ∞ ( c, δ ) = 1.For δ ≥ n →∞ E [1 n,δ n +1 ,δ ] = e c ( e c − ( e c + e δ − e c + δ − e c + e c − e δ + e δ )and l ∞ ( c, δ ) = e c ( e c + e δ − e c + δ − e c + e c − e δ + e δ . We note that l ∞ ( c, δ ) = 1, ∀ c >
0, if δ = 0, which results in the asymptotic independenceof consecutive record indicators in the LDM. Also, there are no critical points for theindex when δ ≥
0. So, in this case l ∞ ( c, δ ) is increasing in c with lim c →∞ l ∞ ( c, δ ) = 1,and decreasing in δ , with lim δ →∞ l ∞ ( c, δ ) = 0, as can be seen in Figure 3. Gatheringthese results, we conclude that l ∞ ( c, δ ) > δ <
0. The asymptoticindependence for records ( δ = 0) was proved in [26]; we have shown here that δ -recordsattract each other for δ < δ > xact and asymptotic properties of δ -records in the linear drift model. - - δ l n ( , δ ) c = = = c l n ( , δ ) δ =- δ =- δ = δ = δ = Figure 3:
Dependence index l ∞ ( c, δ ) for the Gumbel distribution. - - δ l n ( , δ ) n = = = n l n ( , δ ) δ =- δ =- δ = δ = δ = Figure 4:
Dependence index l n (1 , δ ) for the Pareto distribution as functionof δ and n . Let F be the Pareto distribution and c = 1. The probability of δ -record is given insection 4.2. Computations of l n ( c, δ ) are cumbersome and the explicit expression of l n (1 , δ ) can be found in Appendix 10.5.We have lim δ →−∞ l n (1 , δ ) = 1 and lim δ →∞ l n (1 , δ ) = 1 − log(2) ≈ . n >
1. Also, lim n →∞ l n (1 , δ ) = ∞ , for all δ ∈ R , that is, δ -record-attraction growsunboundedly, as n increases. Moreover, it can be proved that l n (1 , δ ) ∼ C n (log n ) as n → ∞ , where C is a constant depending on δ .The sublinear growth of l n (1 , δ ) as n increases can be observed in the right panelof Figure 4, for different values of δ , as well as the decrease in δ . Also, for fixed n (left panel of Figure 4), there is a negative value of δ where the correlation reaches amaximum, as in the Gumbel case. Note that, for negative and small positive values of δ , l n (1 , δ ) >
1, while, for big values of δ , l n (1 , δ ) <
6. Asymptotic behaviour of N n,δ In sections 3 and 4 we have presented properties of the probability that observation n isa δ -record. In this section we analyse the random variable N n,δ , defined as the numberof δ -records among the first n observations, and study its behaviour as n → ∞ .Depending on F , c and δ , it might be the case that only finitely many δ -records are xact and asymptotic properties of δ -records in the linear drift model. N n,δ grows to infinity, we investigate if the ratio N n,δ /n converges (in a certainstochastic sense) to p δ and, in that case, how the oscillations of N n,δ /n around p δ aredistributed.Recall that, in the classical record model ( c = 0), the number of records N n, growsto infinity, and there are universal results ensuring that, for any continuous F , N n, / log n converges to 1, almost surely (a.s.) and ( N n, − log n ) / (log n ) / has, asymptotically, astandard Gaussian distribution. However, when δ (cid:54) = 0, results in [41] and [45] for themodel with c = 0, show that N n,δ may grow to a finite limit and, when it diverges, thecorresponding limit laws depend both on δ and F . We begin by analyzing the situationwhere N n,δ has a finite limit. δ -records Let N ∞ ,δ = lim n →∞ N n,δ be the total number of δ -records along the sequence ( Y n ) n ≥ . Inthis section we find necessary and sufficient conditions for the finiteness of N ∞ ,δ and E [ N ∞ ,δ ].Clearly, these questions are related to the asymptotic behaviour of p n,δ . If p δ > N ∞ ,δ = ∞ . On the other hand, if p δ = 0, it may happen that N n,δ grows sublinearly to ∞ or N ∞ ,δ < ∞ . Since, by Theorem 1, the positivity of p δ is linkedto the finiteness of µ + , we split the analysis in two cases: µ + = ∞ . In this situation, N ∞ ,δ = ∞ a.s. for any c, δ ∈ R .To check this assertion, we first prove that M n := max { Y , . . . , Y n } → ∞ . Observethat µ + = ∞ implies x + = ∞ and ∞ (cid:88) n =1 P [ Y n > a ] = ∞ (cid:88) n =1 P [ X n > a − cn ] = ∞ (cid:88) n =1 (1 − F ( a − cn )) = ∞ , ∀ a ∈ R . (15)From (15) and the second Borel-Cantelli lemma, we conclude that Y n > a infinitelyoften (i.o.), for any a , and so, M n → ∞ , with probability one. This fact clearly implies N ∞ , = ∞ . Now, since, for δ < N ∞ ,δ ≥ N ∞ , , we get N ∞ ,δ = ∞ . On the other hand,for δ >
0, the event { X n + ( c − δ ) n > max ≤ j ≤ n − { X j + ( c − δ ) j }} implies { X n + cn > max ≤ j ≤ n − { X j + cj } + δ } , that is, 1 n, ( c − δ ) ≤ n,δ ( c ). Therefore, N ∞ ,δ ( c ) ≥ N ∞ , ( c − δ ) = ∞ . µ + < ∞ . We distinguish three scenarios depending on the sign of c .If c >
0, we first assume x + − x − > δ − c . In this case, we have p δ > N ∞ ,δ = ∞ is an immediate consequence of the law of large numbers in Theorem 5below. If x + − x − ≤ δ − c , only the first observation will be a δ -record as shown insection 3.1, so N ∞ ,δ = 1. xact and asymptotic properties of δ -records in the linear drift model. c = 0 and δ ≤
0, then N ∞ ,δ = ∞ , since N ∞ ,δ ≥ N ∞ , = ∞ . If c = 0 and δ > N ∞ ,δ < ∞ if and only if (cid:90) ∞ − F ( x + δ )(1 − F ( x )) f ( x ) dx < ∞ , which is also equivalent to E [ N ∞ ,δ ] < ∞ . This is shown in Proposition 7 of the Appendix,by relating this question to the counting process of geometric records, as studied in [45].If c <
0, we proceed as in (15) to obtain ∞ (cid:88) n =1 P [ Y n > a ] = ∞ (cid:88) n =1 P [ X > a − cn ] < ∞ , ∀ a ∈ R , where the last inequality follows from µ + < ∞ . Thus, the first Borel-Cantelli lemmaensures that P [ Y n > a i.o.] = 0, for all a ∈ R , so Y n → −∞ . Then, there exists a randomvariable N < ∞ such that lim n →∞ M n = M N and, consequently, N ∞ ,δ < ∞ . In thiscase, we can also prove that E [ N ∞ ,δ ] < ∞ ; see Proposition 9.Summarizing the above, we give a complete characterization of the (almost sure)finiteness of the number of δ -records in the next theorem. Theorem 3 N ∞ ,δ < ∞ a.s. if and only if one of the following conditions holds(i) c < and µ + < ∞ ,(ii) c = 0 , δ > and (cid:82) ∞ − F ( x + δ )(1 − F ( x )) f ( x ) dx < ∞ ,(iii) c > and x + − x − ≤ δ − c .Moreover, N ∞ ,δ < ∞ a.s. if and only if E [ N ∞ ,δ ] < ∞ . Remark 4
Theorem 3 answers a conjecture posed in [28], stating that the expectednumber of records ( δ = 0 ) in the LDM, with negative trend, remains finite, based onthe observed exponential decay of p n , in a particular case. We have shown that theconjecture holds if only if µ + < ∞ .6.2. Growth of N n,δ to infinity. We now turn our attention to the case N ∞ ,δ = ∞ . More precisely, we are interestedin the convergence of the proportion of δ -records to p δ . For records ( δ = 0) it wasshown in [24] and [25] that N n, /n → p and that oscillations of N n, around p areasymptotically Gaussian.We show here that these results carry over to the case of δ (cid:54) = 0 but leave the prooffor sections 10.3 and 10.4 of the Appendix. As in the aforementioned works, we assume µ + < ∞ and c > x + − x − > δ − c . Note that, by Theorems 1and 3, we have p δ > N ∞ ,δ = ∞ . Theorem 5
Assume µ + < ∞ , c > and x + − x − > δ − c . Then, as n → ∞ ,(a) N n,δ /n → p δ a.s. and E [ N n,δ /n ] → p δ .xact and asymptotic properties of δ -records in the linear drift model. Coefficient Estimate Std.Error p-value β -62.659 18.172 0.00098 β Table 1:
Regression analysis estimations for the temperature data. (b) If, additionally, (cid:82) ∞ x f ( x ) dx < ∞ , then √ n ( N n,δ /n − p δ ) D → N (0 , σ δ ) , where D → stands for convergence in distribution and σ δ is defined in (21) . As it can be seen in the proof of Theorem 5 (a) in the Appendix, the assumptionon independence of the X n can be relaxed to stationary and ergodic and prove that N n,δ /n → E [1 ∗ ,δ ], defined in (20). This is useful because it allows to deal with a widerrange of scenarios, including stationary ARMA processes. Note, however, that E [1 ∗ ,δ ]could differ from p δ in (2).
7. Illustration
We present a practical application of Theorem 5 to a real dataset of temperatures, whereconvergence to the stationary regime is seen for quite small values of n . As pointed outin the introduction, the LDM has been used by [4, 29] to model temperature data in theframework of climate-change.Our dataset consists of means of daily maximum temperatures (in degrees Celsius),for every month of July, from 1951 to 2019, in the city of Zaragoza, Spain. See Figure5 for a data plot. The least squares line fitted to the data (in dotted red), reveals agradual increase of the maximum temperatures over time.For δ -records we choose the value δ = −
1, which is arbitrary and does not respondto any specific reason, other than interpretability of the example. Note that a year willhave a δ -record temperature if the maximum average temperature in July is a record orif it is at a distance smaller than 1 ◦ C from the current maximum. In this framework,we find that 17 out of the 69 observations are δ -records (coloured in red), and 7 of themare records (with circle).The simple linear model for the temperature takes the form T t = β + β t + ε t , (16)where T t is the temperature of year t , and ε t the error term. The results of the least-square estimators of the coefficients and their p -values (assuming Gaussian errors) areshown in Table 1. In addition, we find an adjusted- R of 0 . β = 0 is clearly rejected, using the Student t-test. Moreover, theestimate of β , which represents the average increment of mean maximum temperaturesby year, agrees well with previous estimates of the summer warming trend in Europe,see [4, 29]. xact and asymptotic properties of δ -records in the linear drift model. l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Year T e m pe r a t u r e ( º C ) l l l l l l l l l l l l l l l l l l l l l l l l ll Record d −record, d =−1 Figure 5:
Monthly Mean of Maximum Temperature in July, 1951-2019 inZaragoza (Spain).
Figure 6, along with a p -value greater than 0 . .
58 for the Shapiro-Wilk test applied to the standard regression residuals, indicatethat the hypothesis of independent normal residuals is appropriate.At this point we consider the analogy between the regression model (16) and theLDM. First note that the intercept β is irrelevant when counting δ -records. On theother hand, β has the role of the trend parameter c , whose estimate is ˆ β = 0 . X n in the LDM are represented by the errors ε t , whichwe assume to be zero mean iid. Note that, for applying Theorem 5, there is no need toassume any specific form of the distribution of the X n .Now, since 17 out of 69 observations were identified as δ -records, it is natural toestimate p δ by the empirical record rate, that is,ˆ p δ = n − N n,δ = 17 / ≈ . . Figure 7 illustrates how the empirical δ -record rate evolves with each extra observationand how it seems to stabilize around a constant value, as predicted by Theorem 5(a).Concerning the asymptotic normality (Theorem 5 (b)), we need to estimate thevariance σ δ , defined in (21). To that end we propose the estimator˜ σ δ = ˜ γ n,δ (0) + 2 m (cid:88) k =1 ˜ γ n,δ ( k ) , (17) xact and asymptotic properties of δ -records in the linear drift model. lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll − − Residuals
Year R e s i dua l s −2 −1 0 1 2 − − − QQ−plot
Normal theoretical quantiles S t uden t i z ed R e s i dua l s l l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l − . . . . Lag A C F ACF − . . . . Lag P a r t i a l A C F PCF
Figure 6:
Diagnostic plots of the regression model. Top Left: Residualsvs year. Top Right: Quantile-Quantile of the residual with the normaldistribution. Bottom Left: Autocorrelation Function. Bottom Right: PartialAutocorrelation Function. where m is a given natural number and˜ γ n,δ ( k ) = n − n − k (cid:88) j =1 (1 j,δ − n − N n,δ )(1 j + k,δ − n − N n,δ ) , The estimator in (17) is a version of an estimator proposed in [25], adapted here to dealwith δ -record data. By slightly changing the proof in [25], we can prove convergence of˜ σ δ to σ δ , as n → ∞ (consistency), under the condition m ( n ) = O ( n / ).In order to apply formula (17), we must choose m , of order √ n . In our case, n = 69so we take m = 8, to obtain the estimate ˜ σ δ = 0 . m = 6 ,
7. Therefore, from Theorem 5(b), N n,δ is approximately Gaussian, withmean 17 and variance 23.25 (0 . × times,and compute the value of N ,δ . Figure 8 summarizes the total number of δ -recordsobtained at each of the 10 simulations. The histogram has a Gaussian shape, so theconvergence in Theorem 5(b) to the Gaussian distribution seems to be fast. Moreover, xact and asymptotic properties of δ -records in the linear drift model. . . . . . . n (number of observations) N ( n , d ) / n Figure 7:
Evolution of the δ -record rate for the temperature data. the 0.025 and 0.975 quantiles of the normal distribution N (17 , .
25) are, respectively,7.54 and 26.45. The 0.025 and 0.975 empirical quantiles from the simulated data are 8and 26, showing an excellent fit to the theoretical (asymptotic) distribution.As a conclusion, we see that empirical results and theory are in very close agreement.This means that, even with a small sample, the approximations in Theorem 5 are good,at least for the model considered.
8. Concluding remarks
In this paper we have studied the behaviour of δ -records in the LDM. We have analysedthe asymptotic probability of δ -records, the dependence between δ -record events andthe limiting distribution of the number of δ -records among the first n observations.The behaviour of the asymptotic probability of δ -records shows similarities withthe case of records ( δ = 0); for instance, for positive c , p δ ( c ) > µ + < ∞ ,regardless the value of δ (except for the trivial case δ ≥ x + − x − + c , where no δ -recordsare observed). We also find that p δ ( c ) is a continuous function of δ for every c , while,as a function of c , it is continuous for every c (cid:54) = 0, and a discontinuity arises at c = 0, if x + < ∞ and δ <
0. This differs from records where p ( c ) is a continuous function of c .We have described in detail the probability of δ -record in two examples. For theGumbel distribution, an explicit expression for p δ ( c ) is found, showing that it decreaseswith δ , as a logistic function of − δ . For the cases studied in the Dagum family ofdistributions, we have p δ ( c ) = 0, for every δ, c , since µ + = ∞ . For this family, weinvestigate if the speed of convergence of p n,δ ( c ) to 0, as n → ∞ , depends on δ or xact and asymptotic properties of δ -records in the linear drift model. Number of d −records, d =−1 R e l a t i v e F r equen cy . . . . . Figure 8:
Histogram of the total number of δ -records for the adjustedregression model (10 iterations of 69 observations). not. Since random variables X n , with µ + = ∞ , may produce large values, whichprovoke abrupt changes in record values, we can expect that δ values close to 0 havenegligible impact and so, p n,δ ( c ) /p n, ( c ) →
1. This happens in the case c = 0, where thenumber of δ -records grows at the same speed as the number of records, when the X n are heavy-tailed. However, we find that, for some distributions in the Dagum family, p n,δ ( c ) /p n, ( c ) → a (cid:54) = 1.Parameter δ has a clear impact in the qualitative behaviour of correlations of δ -record events. First, the expression of the limiting correlation is different for δ ≥ δ <
0. For the Gumbel distribution, where record indicators are independent [26],dependence appears when δ (cid:54) = 0; in fact, δ -records in this distribution attract each otherfor δ < δ >
0. For distributions with power law tails, itis known, for c >
0, that correlations between records are positive and increase with n ; see [30]. We have studied the Pareto distribution with c = 1, and obtained that,while the correlations are positive (and increasing in n ) for negative, zero and smallpositive values of δ , they are negative for big values of δ . In fact, for each n , the limitingcorrelation index, as δ → ∞ , is 0.3069.Another interesting finding of the paper is about the behaviour of the randomvariable N n,δ ( c ). We completely solve the question of finiteness of N ∞ ,δ ( c ), that is, ifthere is a finite number of δ -records along the infinite sequence of observations. We showthat this cannot happen for c >
0, for any δ (except if the condition x + − x − < δ − c holds). It cannot happen either when c < X n have an infinite right-tail mean. This last fact solves a problem posed in [28], where the xact and asymptotic properties of δ -records in the linear drift model. c > N n,δ ,which grows to infinity. We give a law of large numbers, showing that the ratio N n,δ /n converges to p δ ( c ) and that its asymptotic distribution is Gaussian, finding the explicitexpression of its normalizing constants, which can be estimated from observed data.This result was already known for records and has been applied to different problems,such as athletic records [24, 25] and climate change [4, 29]. We have illustrated thelimiting result for N n,δ with a set of real data of temperatures of the city of Zaragoza(Spain), showing a good agreement between the theoretical asymptotic results and theobserved data in the example. In fact, even for this relatively short series (69 data),the distribution of the number of δ -records is close to the theoretical limiting Gaussiandistribution.Our results open the door to the use of δ -records for statistical applications inthe LDM. It has been shown that δ -records perform better than records in statisticalinference, using trend-free data [41–43], so we expect that their use in the LDM is alsoadvantageous.
9. Acknowledgements
This research was funded by project PIA AFB-170001, Fondecyt grant 1161319 andproject MTM2017-83812-P of MICINN. The authors are members of the research groupModelos Estoc´asticos of DGA. M. Lafuente acknowledges the support by the FPU grant,funded by MECD.We are thankful to professors J. Abaurrea, A.C. Cebri´an and J. As´ın, from theUniversity of Zaragoza (Spain), for kindly providing us with the temperature data usedin this paper.
10. Appendix p δ ( c ) Proposition 6 (cid:81) ∞ i =1 F ( x + ci − δ ) , as a function of c is continuous at c ∈ R \ { } , forevery x ∈ ( x − , x + ) , x (cid:54) = x − + δ − c . Proof.
Let ( c n ) n ≥ be a real sequence converging to c >
0. We show that ∞ (cid:89) i =1 F ( x + c n i − δ ) → ∞ (cid:89) i =1 F ( x + ci − δ ) , (18)as n → ∞ , for fixed x ∈ ( x − , x + ), x (cid:54) = x − + δ − c .Let x ∈ ( x − , x + ) be such that x < x − + δ − c (this can only happen if x − > −∞ and δ − c > F ( x + c − δ ) = 0, so the right-hand side (rhs) of (18) is xact and asymptotic properties of δ -records in the linear drift model. c n → c , F ( x + c n − δ ) = 0, for n large enough, the left-hand side (lhs) of(18) is also 0 and (18) is proved.Let now x > x − + δ − c , then F ( x + ci − δ ) > i ≥
1. Let (cid:15) > x + c − (cid:15) − δ > x − and let n ≥
1, such that | c n − c | < (cid:15) , for all n ≥ n . We have, for n ≥ n , − log F ( x + c n i − δ ) ≤ − log F ( x + ( c − (cid:15) ) i − δ ) . Since x > x − + δ − ( c − (cid:15) ) and µ + < ∞ , we have − (cid:80) ∞ i =1 log F ( x + ( c − (cid:15) ) i − δ ) < ∞ ,so the dominated convergence theorem yields ∞ (cid:88) i =1 log F ( x + c n i − δ ) → ∞ (cid:88) i =1 log F ( x + ci − δ ) , as n → ∞ , so (18) also holds for x > x − + δ − c . Finally, for c <
0, we have (cid:81) ∞ j =1 F ( x + cj − δ ) = 0 , ∀ x ∈ R , since F ( x + cj − δ ) →
0, as j → ∞ . (cid:3) δ -records Proposition 7
Let c = 0 and δ > . The following conditions are equivalent:(a) N ∞ ,δ < ∞ ,(b) E [ N ∞ ,δ ] < ∞ ,(c) (cid:90) ∞ − F ( x + δ )(1 − F ( x )) f ( x ) dx < ∞ . Proof.
It is clear that Y n is a δ -record if and only if e X n > e δ max { e X , . . . , e X n − } . That is, if the n -th observation in the sequence ( e X n ) n ≥ is a geometric record, withparameter k = e δ , according to [45]. In section 2.1.1 of that paper, it is shown that thetotal number of geometric records, in a sequence of iid random variables, with cdf G , isfinite if and only if (cid:90) ∞ − G ( kx )(1 − G ( x )) dG ( x ) < ∞ . (19)Moreover, in section 2.3.4 of that paper, it is shown that (19) is equivalent to thefiniteness of the expectation of the total number of geometric records. Since G ( x ) = F (log( x )), the result is proved. (cid:3) In the rest of the Appendix, we use the operator (cid:87) to denote the maximum. Then,for instance, (cid:87) ni =1 Y i = max { Y , . . . , Y n } . Lemma 8 (i) If c < , x − > −∞ and µ + < ∞ , then E [ N ∞ ,δ ] < ∞ , ∀ δ ∈ R . (ii) Let ˜ X be a random variable with cdf G , and ( ˜ X n ) n ≥ an iid sequence, independentof ˜ X , with common cdf F , such that G ( x ) ≤ F ( x ) , ∀ x . Let ˜ Y n = ˜ X n + cn, n ≥ .Then, if c < , E [ (cid:80) ∞ j =1 { ˜ Y j > ∨ j − i =1 ˜ Y i + δ } ] ≤ E [ N ∞ ,δ ] .xact and asymptotic properties of δ -records in the linear drift model. Proof. ( i ) First we bound p n,δ ( c ) as follows p n,δ ( c ) = (cid:90) ∞−∞ n − (cid:89) j =1 F ( x + cj − δ ) f ( x ) dx = (cid:90) ∞−∞ n − (cid:89) j =1 F ( x + cj − δ )1 { x + c ( n − − δ>x − } f ( x ) dx = (cid:90) ∞ x − − c ( n − δ n − (cid:89) j =1 F ( x + cj − δ ) f ( x ) dx ≤ − F ( x − − c ( n −
1) + δ ) . So, (cid:80) nj =1 p n,δ ≤ (cid:80) nj =1 (1 − F ( x − − c ( j −
1) + δ )) yielding E [ N ∞ ,δ ] ≤ ∞ (cid:88) j =1 (1 − F ( x − − c ( j −
1) + δ )) < ∞ , since µ + < ∞ . ( ii ) It suffices to check that the δ -record probability for the ˜ Y n fulfills E [1 { ˜ Y j > ∨ j − i =1 ˜ Y i + δ } ] = (cid:90) ∞−∞ G ( x + c − δ ) j − (cid:89) i =2 F ( x + ci − δ ) f ( x ) dx ≤ (cid:90) ∞−∞ j − (cid:89) i =1 F ( x + ci − δ ) f ( x ) dx = p j,δ ( c ) . (cid:3) Proposition 9 If c < and µ + < ∞ , then E [ N ∞ ,δ ] < ∞ . Proof.
It suffices to consider δ <
0, since the number of δ -records is decreasing with δ .Also, we take x − = −∞ as, otherwise, the result follows from Lemma 8 ( i ). Moreover,since there exists c ∈ R such that P ( X n + c > >
0, and the number of δ -records isthe same for the sequences Y n = X n + cn and ˜ Y n = X n + cn + c , we assume withoutloss of generality that P ( X n > − δ ) > . Let N = inf { n ∈ N | X n > − δ } , then N is a geometric random variable and N ∞ ,δ = N (cid:88) j =1 j,δ + ∞ (cid:88) j = N +1 j,δ = N (cid:88) j =1 j,δ + ∞ (cid:88) j = N +1 j,δ { X j > } . For j > N , let ˜1 j,δ = 1 { X j > ∨ j − i = N ( X i + c ( i − j )+ δ ) } { X j > } , then1 j,δ { X j > } = 1 { X j > ∨ j − i =1 ( X i + c ( i − j )+ δ ) } { X j > } ≤ ˜1 j,δ . Note that the ˜1 j,δ , defined for j > N , are the δ -record indicators of the sequence { X N , X N +1 { X N +1 > } + c, X N +2 { X N +2 > } + 2 c, . . . } . Now, taking expectations we have E [ N ∞ ,δ ] ≤ P ( X > δ ) + ∞ (cid:88) i =1 E [˜1 i,δ ] < ∞ , since the last sum is bounded by Lemma 8 ( ii ). (cid:3) xact and asymptotic properties of δ -records in the linear drift model. N n,δ ( c )Define the bilateral LDM, as in (1), but letting n ∈ Z instead of n ∈ N . Associated tothis model, define, for n ∈ Z , M ∗ n = max { Y i : i ≤ n } , ∗ n,δ = 1 { Y n >M ∗ n − + δ } , (20)and, for n ∈ N , N ∗ n,δ = n (cid:88) k =1 ∗ k,δ . Theorem c > µ + < ∞ . Then N n,δ ( c ) /n → p δ ( c ) a.s. as n → ∞ . Proof.
It is clear thatlim n →∞ P [ Y n > a ] = lim n →∞ P [ X n > a − cn ] = 1 , ∀ a ∈ R , thus Y n → ∞ and M n → ∞ a.s. Also, since µ + < ∞ , it is known by a Borel-Cantelliargument that M ∗ < ∞ a.s. Gathering these facts, we know that ∃ < N < ∞ a.s.such that 1 ∗ N, = 1 almost surely. From the definition of 1 ∗ n, , given n ∈ N we have1 n, ≥ ∗ n, , and so 1 N, = 1 a.s., entailing M ∗ n = M n and 1 n,δ = 1 ∗ n,δ a.s. ∀ n > N . So, ∞ (cid:88) k = N +1 k,δ = ∞ (cid:88) k = N +1 ∗ k,δ a.s. Also, we know that 1 ∗ n,δ is a strictly stationary and ergodic sequence. Applying Birkhoff’sTheorem we have N ∗ n,δ n = 1 n n (cid:88) k =1 ∗ k,δ → E [1 ∗ ,δ ] a.s. Now, let ( a n ) n ≥ be a real sequence diverging to ∞ . Then (cid:12)(cid:12)(cid:12)(cid:12) N n,δ − N ∗ n,δ a n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Na n (cid:12)(cid:12)(cid:12)(cid:12) → a.s. since N does not depend on n . Finally, since (cid:12)(cid:12)(cid:12) N n,δ − N ∗ n,δ n (cid:12)(cid:12)(cid:12) → a.s. and N ∗ n,δ n → E [1 ∗ ,δ ] a.s. ,we have N n,δ n → E [1 ∗ ,δ ] a.s. Finally, E [1 ∗ ,δ ] can be written as the rhs in (2), yielding E [1 ∗ ,δ ] = p δ ( c ). N n,δ ( c )A proof of Gaussian convergence for the number of δ -records, based on the ideas in [24],is not straightforward. The main problem arises when considering the joint probabilityof two observations being δ -records. While in the case of records this quantity can beexplicitly written as follows E [1 i, i + m, ] = (cid:90) ∞−∞ i − (cid:89) k =1 F ( y + ck ) (cid:90) ∞ y − cm m − (cid:89) j =1 F ( s + cj ) f ( s ) dsf ( y ) dy, xact and asymptotic properties of δ -records in the linear drift model. δ (cid:54) = 0 there is no such analytical expression. In order to solve this problemwe introduce the following general bounds, which do not depend on the specification ofthe model for the sequence ( Y n ) n ≥ . Proposition 10
Let ( Y k ) k ∈ Z be a sequence of random variables and consider the events A = (cid:110) i − (cid:87) k = −∞ Y k + δ < Y i (cid:111) , B= (cid:110) i + m − (cid:87) k = i +1 Y k + δ < Y i + m (cid:111) , C = { Y i − δ < Y i + m } and E = { Y i + δ < Y i + m } . Then, if δ ≤ ,a1) P [ A ∩ B ∩ C ] ≤ E [1 ∗ i,δ ∗ i + m,δ ] anda2) P [ A ∩ B ∩ E ] ≥ E [1 ∗ i,δ ∗ i + m,δ ] .Also, if δ ≥ ,b) P [ A ∩ B ∩ E ] = E [1 ∗ i,δ ∗ i + m,δ ] . Proof . a1) Note that 1 ∗ j,δ is the indicator of D j = (cid:110) j − (cid:87) k = −∞ Y k + δ < Y j (cid:111) , j = i, i + m .Then we must show that A ∩ B ∩ C ⊆ D i ∩ D i + m .First, it is clear that A = D i . Also, observe that C ⊆ E and that A ∩ C ⊆ (cid:110) i − (cid:87) k = −∞ Y k + δ < Y i + m (cid:111) , since δ ≤
0. From the inclusions above we have A ∩ B ∩ C ⊆ (cid:110) i − (cid:95) k = −∞ Y k + δ < Y i + m (cid:111) ∩ E ∩ B = D i + m and the conclusion follows. a2) Trivial. b) It is clear that D i ∩ D i + m ⊆ A ∩ B ∩ E and that A ∩ B ∩ E ⊆ D i , because A = D i . Also, since δ ≥
0, we have A ∩ E ⊆ (cid:110) i − (cid:87) k = −∞ Y k + δ < Y i + m (cid:111) , so A ∩ B ∩ E ⊆ (cid:110) i − (cid:95) k = −∞ Y k + δ < Y i + m (cid:111) ∩ E ∩ B = D i + m , which completes the proof. (cid:3) Note that, although it is unnecessary in our setting, the reverse a1) inequality alsoholds for δ ≥
0. Under the assumptions of the LDM, the lhs of the first two bounds inthe previous proposition have analytical expressions. The strategy to prove Gaussianconvergence is to work with the corresponding bounds of E [1 i,δ i + m,δ ], which are shownto be tight enough to achieve our purpose. So, with this result we slightly modify thenecessary bounds and rebuild the martingale approach in [24], to prove convergence tothe Gaussian distribution. Theorem (cid:82) ∞ x f ( x ) dx < ∞ and let c > , δ ∈ R , such that p δ >
0. Then, as n → ∞ , xact and asymptotic properties of δ -records in the linear drift model. √ n ( n − N n,δ − p δ ( c )) D → N (0 , σ δ ( c )) , where σ δ = p δ − p δ + 2 ∞ (cid:88) m =1 ( E [1 ∗ i,δ ∗ i + m,δ ] − p δ ) . (21) Proof.
For simplicity, we only consider the case δ ≤ δ > − δ < x + as, otherwise, we can define X (cid:48) n = X n + ( − δ − x + ), n ≥
1; the number of δ -records in both models is the same and − δ < x (cid:48) + , where x (cid:48) + isthe right-end point of X (cid:48) n .The proof is split into several steps. We claim that0 ≤ p n,δ − p δ ≤ c − (cid:90) ∞ c ( n − / − δ (1 − F ( s )) ds + F ( − δ ) (cid:98) ( n − / (cid:99) . (22)The first inequality follows from p n,δ − p δ = (cid:90) ∞−∞ (cid:32) n − (cid:89) j =1 F ( y + cj − δ ) − ∞ (cid:89) j =1 F ( y + cj − δ ) (cid:33) f ( y ) dy ≥ . For the second, let u = n − (cid:81) j =1 F ( y + cj − δ ) and v = ∞ (cid:81) j =1 F ( y + cj − δ ). Then, from theelementary inequality u − v ≤ u − uv , we have p n,δ − p δ ≤ (cid:90) ∞−∞ u (1 − v ) f ( y ) dy. (23)The integral in the rhs of (23) is split into two terms A, B , that we bound. Let A = (cid:82) − c ( n − / −∞ u (1 − v ) f ( y ) dy and B = (cid:82) ∞− c ( n − / u (1 − v ) f ( y ) dy , then A ≤ (cid:90) − c ( n − / −∞ n − (cid:89) j =1 F ( − c ( n − / cj − δ ) f ( y ) dy ≤ n − (cid:89) j =1 F ( c ( j − ( n − / − δ ) ≤ (cid:98) ( n − / (cid:99) (cid:89) j =1 F ( c ( j − ( n − / − δ ) ≤ (cid:98) ( n − / (cid:99) (cid:89) j =1 F ( − δ ) = F ( − δ ) (cid:98) ( n − / (cid:99) . (24) xact and asymptotic properties of δ -records in the linear drift model. B we have B ≤ (cid:90) ∞− c ( n − / (cid:32) − ∞ (cid:89) j = n F ( y + cj − δ ) (cid:33) f ( y ) dy ≤ (cid:90) ∞− c ( n − / ∞ (cid:88) j = n (1 − F ( y + cj − δ )) f ( y ) dy ≤ (cid:90) ∞− c ( n − / (cid:18)(cid:90) ∞ z = n − (1 − F ( y + cz − δ )) dz (cid:19) f ( y ) dy ≤ (cid:90) ∞− c ( n − / (cid:18) c − (cid:90) ∞− c ( n − / c ( n − − δ (1 − F ( s )) ds (cid:19) f ( y ) dy ≤ c − (cid:90) ∞ c ( n − / − δ (1 − F ( s )) ds. (25)So, from (24) and (25), (22) holds. Let r m,δ = E [1 ∗ i,δ ∗ i + m,δ ], which is well defined since it does not depend on i . Webound r m,δ by applying Proposition 10 as follows: r m,δ = P [ Y i , Y i + m are δ -records]= P (cid:34) Y i > (cid:95) l (cid:95) l (cid:95) l m − (cid:95) l =1 Y i + l + δ, Y i + m > Y i + δ (cid:35) = (cid:90) (cid:90) y
2, we use (13) for δ < δ ≥ . Let a = n − δ , A = ( δ − δ (1 − a ) + ( n −
1) log a )( n log( a + 1) − δa ) ,B = − (cid:0) δ ( n −
2) + δ − n − δ ( n −
2) + δ ( n − n + 5) n + n + 1 (cid:1) log( a + 1) ,C = ( a −
1) log( a +1 − δ ) − ( δ − a (cid:0) δ ( a − − ( n − a log(4 a ) (cid:1) +(1 − a ) log(( a − δ +1)( a +1)) . Then, if δ < l n (1 , δ ) = B + CA . Let a = n − δ , A = ( δ − ( δ − a )( δ (1 − a ) + ( n −
1) log a )( − δa + n log( a + 1)), B = ( a − δ ) (cid:18) ( δ − δ ( a −
1) + ( δ − n −
1) log (cid:18) a − δ + 1(2 − δ ) a (cid:19)(cid:19) ,C = − log(2 − δ )( δ ( δ + 2) − δn + n − (cid:1) + ( δ − ( n −
1) log( a − δ + 1) . Then, if 0 < δ < l n (1 , δ ) = a ( B + C ) A . If δ = 1, l n (1 ,
1) = ( n − (( n − n − n −
1) log( n − n − − n + ( n −
1) log( n −
1) + 2)( − n + n log( n ) + 1) . Finally, let a = n − δ , A = ( δ +log δ − n log δ − n +( n −
1) log( n − δ − ( δ − a ), A = ( δ − n log δ − n + n log n ) ,B = (log δ ) (cid:0) δ ( δ + 2 δ −
1) + (2 δ − n − δ n + n (cid:1) ,C = ( δ − ( n −
1) log( n − − ( a − (cid:0) ( δ − δ − a ) + (2 δ −
1) log(2 δ − a − (cid:1) . Then, if δ >
1, and δ / ∈ { n/ , n, n + 1 } (otherwise, the values of the index are given bythe continuous extension at these points), l n (1 , δ ) = a ( B + C ) A A . xact and asymptotic properties of δ -records in the linear drift model.
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