On rare and extreme events

On rare and extreme events

Thierry Huillet a,b,*, Henri-Francois Raynaud c

a LIMHP-CNRS, Universit�e Paris XIII, Institut Galil�ee, 93430 Villetaneuse, Franceb LPTM, Universit�e de Cergy-Pontoise, Neuville-Sur-Oise, 95031 Cergy-Pontoise, France

c L2TI, Universit�e Paris XIII, Institut Galil�ee, 93430 Villetaneuse, France

Received 10 October 1999; received in revised form 14 January 2000; accepted 1 March 2000

Communicated by G. Nicolis

Abstract

In many ®elds of applied physics, the space-time phenomena to be studied may be described in the following way: events of random

amplitudes occur randomly in time. We investigate some statistical properties of this model, with special emphasis on situations where

the model for the waiting time between consecutive events or the amplitude of individual events are fractal (power-law) distributions

with in®nite mean value (the rareness or extreme event hypothesis). Limit laws for cumulative partial sums and the extremal process are

characterized. Using asymptotical results on backward and forward recurrence times, limit laws are investigated for the physically

realistic situation when the cumulative process is only observed starting from some non-zero observational time. Ó 2001 Elsevier

Science Ltd. All rights reserved.

1. Introduction

As applied scientists, we are faced with many natural phenomena appearing to us as a sequence in timeof inherently irregular data in space: in Hydrology, this could be the sequence of rainfalls at some location,in Geophysics, a random energy release's chronicle of individual earthquakes in some country. It could alsobe the sequence of damages met by the customers of some insurance company in Finance, or the randomusers' demands for network or energy resources in Telecommunications' or power supply managementtechnology.

In this article, we are interested by a ``renewal type'' model for which such random physical phenomenaare to be described in the following simple way: events of random independent and identically distributed(i.i.d.) positive magnitudes occur at random times, the inter-arrival times (holding or waiting times) ofwhich form an i.i.d. sequence: for such sequences, both instants of occurrence and amplitudes of the eventsare of concern. Special attention is paid in this article to the possibility that both magnitudes and waitingtimes' sequences are heavy-tailed with tail exponents a > 0 and d > 0, possibly smaller than one (the in®nitemean value hypothesis). Such processes endeavour special statistical properties. To take an image fromClimatology, storms in arid regions are reputed to occur rarely but with extreme violence. Reports onheavy-tailed amplitudes with exponent a � 2=3 < 1 also exist in the context of earthquake magnitude data[18], in accordance with the Gutenberg±Richter theory [13]. In a similar context, several articles presentedempirical evidence of heavy-tailed holding times between consecutive earthquakes [24,31]. A comparativestudy with ®nite mean values situations is therefore supplied.

www.elsevier.nl/locate/chaos

Chaos, Solitons and Fractals 12 (2001) 823±844

* Corresponding author.

E-mail address: [email protected] (T. Huillet).

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 4 6 - 1

In Section 2, we brie¯y recall standard de®nitions and notations for renewal processes, together withsome well-known properties. Section 3 focuses on the cumulated magnitudes' partial sums. More pre-cisely, we exhibit all limit laws for suitably rescaled versions of these partial sums observed from theorigin of time. In the context of rare and extreme events, these limit laws are identi®ed as fractal-typedistributions, such as L�evy, inverse L�evy and other related laws. These asymptotic results are available inthe literature, albeit in the form of scattered material. They are presented here in a simple and self-contained way, using tools which turn out to be relevant for investigating other properties of the model,such as forward and backward recurrence times. Whereas the heavy-tailed amplitudes hypothesis tends toaccelerate the partial sums growth with time, heavy-tailed waiting times tend to slow this process, with acompetition between the two. As the tools to be employed are quite similar, we also give, in Section 4 ananswer to the ``time between failure question'' which is important in practice: how long should one waitbefore the ®rst event of magnitude exceeding some (possibly high) threshold is achieved; it turns out thatthe answer is strongly dependent on the ®niteness or not of the expected time between consecutive eventsand the asymptotic behavior of this variable is supplied in the various cases envisaged. These constituteimportant issues in the prediction of ¯oods, catastrophic earthquakes, ruin or bu�er over¯ows in theexamples just mentioned.

In some instances, however, real processes of interest are only observed starting from time t > 0 andonwards, over a span of time s > 0; in such situations, no data are available between time zero (assumedhowever to be known) and time t > 0. What conclusions may now be drawn from these observations,especially if the starting observational time t is ``large''? To answer this question, a particular study of theincrements process is needed; to that purpose, a comprehensive study of the variation of the partial sumsprocess between times t and t � s is supplied in Section 6 underlining the important question of stationarity(or lack thereof) of the increments of the considered model.

It happens that this study strongly rests on the asymptotic properties of the ``forward recurrence time''which is the time separating current observational time t from the time of occurrence of the next event in thesequence. We therefore focus on the asymptotic behavior of forward (and backward) recurrence time underthe hypothesis that the expected values of individual magnitudes and waiting times are or not ®nite, withvery distinct asymptotic behaviors in each case; as these random variables are of particular practical interestin their own right, the whole of Section 5 is devoted to them.

The impact of this study on the understanding of the increments process is next described in some detailin Section 6. The conclusions may be summarized as follows. If the expected waiting time is ®nite, when thestarting observational time t is large, the increments process on intervals of length s exhibit di�erent large sasymptotic regimes depending on whether the variance of the waiting time is ®nite or in®nite. Under thesame conditions for t, if the expected holding time is in®nite, the length s of the interval over which theincrements process should be evaluated should be proportional to t: s � ts0, in order to capture enoughevents to compensate the rare event hypothesis; large s0 asymptotic regimes can then be obtained.

2. Renewal processes: de®nitions and notations

We ®rst recall salient facts arising from the modelling of events occurring randomly in time. This is areview of classical material [2±30], whose purpose is to introduce the de®nitions and notations used throughthis article.

2.1. The elementary counting process

Suppose at time t � 0, some event occurs for the ®rst time. Suppose successive events occur in the futurein such a way that the waiting times between consecutive events form an i.i.d. sequence �T ; Tm; m P 1�, with

Tm �d T ; m P 1; �2:1�waiting time T is assumed to have a density function (d.f.), say fT �t�. We shall also need the probabilitydistribution function (p.d.f.) of T and its complement to one (c.p.d.f.), i.e.,

824 T. Huillet, H.-Fr. Raynaud / Chaos, Solitons and Fractals 12 (2001) 823±844

FT �t� :� P�T 6 t�; �2:2�F T �t� :� 1ÿ FT �t�: �2:3�

We are then left with a sequence of events occurring at times

T 0 � 0; T n :�Xn

m�1

Tm; n P 1: �2:4�

Let N�t�; t > 0, count the random number of events which occurred in the time interval �0; t�. Clearly,

N�t� �Xn P 0

1�T n6 t� �2:5�

with 1�� the set indicator function which takes the value one if the event is realized, zero, otherwise.As a result, an essential feature of such processes is that the events ``N�t� > n'' and ``T n6 t'' coincide.Such random processes are called pure counting renewal processes (the adjective pure is relative to the

hypothesis which has been made that the origin of time is an instant at which some event occurred; if thisnot the case, the adjective delayed is currently employed and the ®rst event occurs at time T 0 :� T0 > 0,independent of �T ; Tm; m P 1� but not necessarily with the same distribution). If in additionR�1

0fT �s� ds � 1 (T is ``proper'') such renewal processes are said to be recurrent; this has to be opposed to

transient renewal processes for whichR�1

0fT �t� dt < 1, corresponding to ``defective'' T ; allowing for a ®nite

probability that the ®rst event never occurs, i.e., occurs at time t � �1. In the sequel, we shall avoidtransient processes and limit ourselves to recurrent ones. However, among recurrent processes, we shalldistinguish between positive recurrent processes for which the average renewal time ET :� h < �1 and nullrecurrent processes for which ET � �1.

If ET � �1, we shall limit ourselves to situations where this occurs as a result of ``heavy-tailedness'' ofthe waiting time

F T �t� �t"�1

cdtÿd �2:6�

with d 2 �0; 1�. Here, cd > 0 is a scale factor for T. In other words cd � td0 for some t0 > 0 ®xing the time-scale itself. This terminology deserves the following comment.

Remark 1. Assume for simplicity a ``true'' power-law model for T, i.e., for t > t0,

F T �t� � tt0

� �ÿa

�2:7�

with t0 > 0 and a > 0. Its moment generating function can be computed in closed form, and is

ET b � aaÿ b

tb0 �2:8�

for b < a. As a consequence, this distribution has in®nite mean value when a < 1. However, one can alwayscompute its median value, which is easily found to be t021=a. Hence, t0 should be interpreted as a centralityparameter, i.e., as a scale factor. On the other hand, it would be misleading to interpret t0 as a cut-o� value.To grasp this, let us select some cut-o� value tc > t0, and let ~Tc :� �T ÿ tc�1�T > tc� > 0 denote the excessvariable T ÿ tc given T > tc. Its c.p.d.f. F ~Tc

�t� is equal to

F ~Tc�t� � F T �t � tc�

F T �t�� 1

�� t

tc

�ÿa

; t > 0: �2:9�

This quantity is independent of the characteristic scale t0 of T: thus, the intrinsic scale of a truncated power-law distribution (de®ned as the median value of ~Tc, i.e., tc�21=a ÿ 1�) is completely determined by the cut-o�threshold, and not by the scale factor of the original variable.

T. Huillet, H.-Fr. Raynaud / Chaos, Solitons and Fractals 12 (2001) 823±844 825

Finally, we note that if the holding times �T ; Tm; m P 1� are exponentially distributed, this countingprocess boils down into the familiar Poisson process.

We shall call K�t� :� EN�t� the intensity of the pure renewal process and a�t� :� dK�t�=dt its rate (i.e., theinstantaneous frequency at which events occur at time t); the function K�t�=t is called the frequency of thephenomenon. Note that the rate and frequency functions have the same asymptotic behavior for largetimes.

By studying their Laplace transforms, the asymptotic behaviors of K�t� and a�t� are easily obtained usingTauberian theorems [2,14]. These results are summarized in Table 1. A salient feature of the case h � �1 isthat the intensity goes to in®nity more slowly than t, and that the rate function tends to zero algebraically.As time goes to in®nity, the events get sparser and sparser, owing to the in®nite average hypothesis of thewaiting times.

2.2. Compound recurrent renewal processes and the cumulative magnitude

The process N�t� counts the number of events which occurred before time t. Assume now some physicalphenomenon to be described by a compound renewal process: events of random i.i.d. magnitudes, say�X ;Xm; m P 1�, occur at random times T n; n P 1, the waiting times of which forming an i.i.d. sequence.

In the sequel, we shall note fX , FX and F X the d.f., p.d.f. and c.p.d.f. of the local magnitude X.This model was introduced in Physics in [25], and its properties are extensively examined in the context

of continuous-time random walk (CTRW) models, including under the rareness hypothesis, [19±33], in thefractal-time random walk (FTRW) model. One may be interested by a process which cumulates thisrandom number of random amplitudes. A compound renewal process is to the counting renewal processwhat a compound Poisson process is to a Poisson process itself. Physical situations where the relevance ofthis model holds are numerous: think of the random magnitudes as a claims' sequence in insurance risktheory, as the energy release of individual earthquakes in geophysics or as random water inputs ¯owing intoa dam in hydrology. Summing the individual contributions yields the total claim amount (cumulativeenergy release and water input) over a certain lapse of time. In all these applications we have in mind, themagnitude X is a positive random variable; we shall therefore only deal with this case in the sequel.

Let then the cumulative magnitude X �t� be de®ned by

X �t� � x�0� �XN�t�m�1

Xm; �2:10�

where �X ;Xm; m P 1� is the i.i.d. random magnitudes' sequence and x�0� is the initial deterministic state,which in the sequel shall be set to zero. The distribution of this variable may be derived by using thefollowing identity in distribution:

X �t� �d 0 � 1�T > t� � X �T �ÿ � X �t ÿ T �� 1�T 6 t�; �2:11�where T > 0 is a ``proper'' positive random variable known as the ®rst renewal time of X �t�. Such processesare called compound pure recurrent renewal processes.

Let us brie¯y comment this formula. At time T, X �t� undergoes a ®rst (random) jump with amplitudeX �T � > 0, possibly dependent on the occurrence time T of this jump.

Let us now freeze the time t at which X �t� is to be evaluated. If the realization of time T exceeds the time tof interest, the process X �t� is in the deterministic initial state x�0� � 0. If T � s6 t, the value of X �t� is theindependent sum of the ®rst jump of amplitude X �s� plus a statistical copy of the process X �� in the

Table 1

Asymptotic behavior of intensity and rate functions when t " �1ET � h < �1 F T �t� �

t"�1cdtÿd, d 2 �0; 1�

Intensity K�t� � EN�t� t=h td=cd

Rate a�t� � dK�t�=dt 1=h tdÿ1=cd


remaining time t ÿ s, conditionally to the event T � s. Renewal processes generalize the familiar compoundPoisson process family in that the waiting time distributions between spikes is an i.i.d. sequence, albeit notnecessarily exponentially distributed.

Let us now translate the de®nition (2.11) in terms of the evolution of the Laplace transform of X �t�,following [32]. Let

UX �t; k� :� EeÿkX �t�; �2:12�/X �s; k� :� EeÿkX �s�; �2:13�

respectively, stand for the Laplace transforms of the cumulative process X �t� and of a local magnitude X �s�which occurred at time s6 t. Then

UX �t; k� � P�T > t� �Z t

0

UX �t ÿ s; k�/X �s; k�fT �s� ds: �2:14�

We shall now make an additional simplifying hypothesis.Assume that the local magnitudes are independent of their occurrence time (the decoupling hypothesis in

the CTRW model); then /X �s; k� � /X �k� and the Laplace transform of the conditional magnitude X isindependent of the particular realization s of the occurrence time T. Then (2.14) boils down to

UX �t; k� � P�T > t� � /X �k�Z t

0

UX �t ÿ s; k�fT �s� ds: �2:15�

This is the integral (convolution) equation that UX �t; k� now satis®es. Using the convolution equation(2.15), the Laplace transform of UX ��; k� veri®es

UX �p; k� �1ÿ /T �p�

p 1ÿ /T �p�/X �k�� ; �2:16�

provided that /T �p�/X �k� < 1. Thus, the solutions of /T �p� � /X �k�ÿ1are the poles of UX �p; k�.

Processes obeying Eq. (2.16) are known as pure renewal processes with stationary local magnitudes [15].

Remark 2. Renewal process are more general than standard processes with stationary independent in-crements (sii), such as Poisson, because they are not Markovian as the integral equation (2.14) shows: thedistribution at time t of X �t� depends (in general) on the distribution of X �s�, s < t. However, it can easilybe shown that they include the compound Poisson class (a very important sub-class of processes with sii)which may be recovered if the renewal time T is assumed to be exponentially distributed, because of thememory-less character of the exponential distribution.

3. Asymptotics of the cumulative partial sums

The main purpose of this section is to sort out the asymptotic behavior of solutions of classical renewalproblems in the presence of power-law distributions. More precisely, we are concerned with the behaviorfor long times of the cumulated variable X �t�. It turns out that the asymptotics of this variable is stronglydependent on whether the mean value h between consecutive events is ®nite or in®nite; in each of these sub-cases, it seems important to distinguish between situations where the mean value, say l, of the local am-plitude X is itself ®nite or in®nite. We shall recall very brie¯y the results for the simple case where both hand l are ®nite, and focus instead on the asymptotic behaviors associated with fractal power-law typedistributions for waiting time or amplitude. While these results are not really new, they are derived herefrom the renewal equations (2.15) and (2.16) in a simple and essentially self-contained manner. The resultspresented in this section are summarized in Table 2.

If EX � �1, we shall limit ourselves to situations where this occurs as a result of ``heavy-tailedness'' ofthe amplitude X:


F X �x� �x"�1

caxÿa �3:1�

with a 2 �0; 1�. Here, ca > 0 is a scale factor for X. In other words ca � xa0 for some x0 > 0 ®xing the space-

scale itself. Let us now discuss the four possible combinations.

3.1. h < �1 and l < �1

The main result in this case is the convergence in probability

1

�t=h�X �t�l!Pr

t"�11 �3:2�

which is consistent with the law of large numbers.

Sketch of Proof. Following [8], observe that in the case h < �1:

UX �t; k� �t"�1 expÿtsX �k�; �3:3�

where the ``free energy'' function sX �k� is implicitly de®ned by

/T �ÿsX �k�� /X �k�ÿ1: �3:4�

When the mean value l :� EX of the local magnitude X is ®nite, /X �k� �k"0�

1ÿ lk; as a result, sX �k� tends tozero as k tends to zero, and

/T �ÿsX �k�� k"0�

1� hsX �k� � 1� lk: �3:5�

We thus have

sX �k� �k"0�

lh

k; �3:6�

and

Eeÿk�1=t�X �t� � UX �t; k=t� �t"�1

expÿ lh

k: �3:7�

Limit theorems in distribution are known to exist in this case for the centered rescaled variable�X �t�=lÿ t=h�=C�t� for some increasing normalizing function depending on the particular distribution of�T ;X � [20].

3.2. h < �1 and l � �1

The main result is

1

�jat=h�1=aX �t� !d

t"�1La; �3:8�

Table 2

Asymptotic behavior of the cumulative partial sums when t " �1ET � h < �1 F T �t� �

t"�1cdtÿd; d 2 �0; 1�

EX � l < �1 tÿ1X �t�!Prl=h tÿdX �t�!d �l=jd�Id

F X �x� �x"�1

caxÿa; a 2 �0; 1� tÿ1=aX �t�!d �ja=h�1=aLa tÿd=aX �t�!d jd;aLd;a


where La > 0 is an asymmetric L�evy variable, whose Laplace transform is

exp �ÿC�1ÿ a�ka�: �3:9�Note that as a < 1, the cumulated variable X �t� grows faster than time t, as a result of the heavy-tailedcharacter of the magnitude X.

Sketch of Proof. From our power-law assumption, /X �k� �k"0�

1ÿ caka. Thus, the function sX �k� now tends to

zero as k tends to zero in the following way

sX �k� �k"0�

ca

hka: �3:10�

This shows that

Eeÿk�1=t1=a�X �t� � UX t; k=t1=aÿ � �

t"�1expÿ ca

hka: �3:11�

Thus, we found the convergence in distribution to an asymmetric L�evy variable Sa�r; 1; 0� in the termi-nology of [26, page 11], with the scale parameter r > 0, related to ca through ca � Cara. Here,Ca � �1ÿ a�=�C�2ÿ a� cos�pa=2�� (see [26, page 17]). The constant which appears in (3.8) is

ja � ca=�aCa�: �3:12�

3.3. h � �1 and l < �1

The case h � �1 is a ``rare event'' hypothesis in time, as the expected time between consecutive events isin®nite. In this case, the asymptotics is of a very di�erent nature.

The main result is

jd

tdX �t�l!d

t"�1Id; �3:13�

where Id is the ``inverse'' L�evy variable obtained while raising a L�evy variable of index d, say Ld, to thepower ÿd: Id � Lÿd

d .From this equation, as d < 1, the cumulated variable X �t� grows slower than time t, as a result of the

heavy-tailed character of the waiting time T: the time between consecutive events is long and this propertyslows the speed of growth of X �t�.

Sketch of Proof. From the heavy-tailed character of the waiting times, we have upon scaling (see [14, XI,(5.6), page 373])

jd

tdN�t� !d

t"�1Id; �3:14�

where jd � cd=�dCd�. As N�t� goes to in®nity with probability one, we get from the law of large numbers

1

N�t�X �t�l!Pr

t"�11: �3:15�

Thus, combining the two results (3.14) and (3.15), we get the announced result (see also [27, page 27]).Another way to look at this result is the following: consider the random variable Id�t� � �l=jd�tdId. It has

the Mittag±Le�er function

Ud�t; k� �Xn P 0

1

C�1� nd� �ÿkl=jd�ntnd; �3:16�


as Laplace transform in space (see [14,XIII.8, (8.4), page 453]), so that the Laplace transform of Ud��; k� is

Ud�p; k� � 1

p 1� l=cd�pÿdk�� : �3:17�

Now, under our rare and extreme events hypothesis, /X �k� �k"0�

1ÿ lk, and /T �p� �p"0�

1ÿ cdpd. Thus (2.16)yields

UX �p; k� ��k;p�"0�Ud�p; k� �3:18�

proving (3.13).

3.4. h � �1 and l � �1

The main result is (see also [20, page 27])

1

jd;atd=aX �t� !d

t"�1Ld;a; �3:19�

where the random variable Ld;ais obtained as the ratio Ld;a � La=�Ld�d=a � �Id=Ia�1=a, with �La; Ld�, (respec-tively, �Ia; Id�) two independent L�evy (respectively, inverse L�evy) variables with respective parameter aand d.

From this result, we may now observe a competition between the heavy-tailed character of the mag-nitude X which forces the cumulated variable X �t� to grow faster than time t and the heavy-tailed characterof the times T between consecutive events which slows its growth. A critical situation is when d � a, inwhich case, X �t� grows like time t and the two e�ects exactly compensate, and (3.19) reduces to

1

tX �t� !d

t"�1La;a; �3:20�

where La;a is the ratio of two independent L�evy variables with the same parameter a.

Sketch of Proof. Note that, under our hypothesis: /X �k� �k"0�

1ÿ caka, and /T �p� �

p"0�1ÿ cdpd. Proceeding as in

the previous case, we obtain from (2.16):

UX �p; k� ��k;p�"0�bUd;a�p; k� �3:21�

with

Ud;a�p; k� � 1

p 1� ca=cd�pÿdka�� : �3:22�

This time, Ud;a�p; k� is the space-time Laplace transform of the random variable Ld;a�t� � jd;atd=aLd;a,proving the result (3.19).

The constant jd;a which appears may be expressed as

jd;a � �ja=jd�1=a � dcaCd

acdCa

� �1=a

; �3:23�

in terms of the scaling constants ca and cd already encountered.

3.5. Statistical interpretation of the limit results

A coarse analysis of data could be performed in the following way. Consider the empirical log±log plotsof N�t� and X �t� against time t, say n�t� and x�t�.


· h < �1 : A particular version of (3.2) is

1

�t=h�N�t�!Pr

1: �3:24�

From this equation, the empirical evidence of h < �1 is

log n�t�log t

!t"�1

1: �3:25�

In other words, the log±log plot of n�t� against thas for asymptote for large values of t a straight line withslope one. Two sub-cases then may arise:(1) l < �1: the empirical evidence of this fact is, from (3.2)

log x�t�log t

!t"�1

1; �3:26�

(2) l � �1: this happens if and only if, from (3.8)

log x�t�log t

!t"�1

1=a > 1: �3:27�

· h � �1 : if this property holds, then, from (3.14)

log n�t�log t

!t"�1

d < 1: �3:28�

Two sub-cases then may also arise:(1) l < �1: this may be checked, from (3.13), by

log x�t�log t

!t"�1

d < 1: �3:29�

(2) l � �1: empirical evidence of this property is, from (3.19)

log x�t�log t

!t"�1

d=a > d: �3:30�

Thus the empirical log±log plots of both n�t� and x�t� against time t allow to decide between the fourstatistical situations envisaged and sometimes to derive an estimator of both tail exponents d and a, simplyby inspection of the log±log slopes in the large time limit.In fact, a more re®ned ``test'' could proceed in this way. Models (3.8), (3.13) and (3.19) all express in theform

1

C�t�X �t� !d

t"�1L �3:31�

for some normalizing function C�t� and some limit variable L. Choose a particular model (®xing C�t�and L). Fix a small real number, say q � 0:05. In each case, it is possible to compute the number �t�q�de®ned by PfX �t� > �t�q�g � q. The number �t�q� is therefore the value which X �t� is unlikely (at level q) toexceed. From (3.8), (3.13) and (3.19), �t�q� can be e�ectively computed from inspection of the tails of L ineach case through

P Lf > �t�q�=C�t�g � q: �3:32�This construction therefore yields an approximation of the width of the con®dence interval of X �t�. Tosummarize, the test works as follows: (a) select a model and evaluate the empirical value, say x�t� of X �t�;(b) select some small q, and determine the level value �t�q� for which (3.32) holds; (c) if x�t� > �t�q�, rejectthe hypothesis that X �t� ®ts the selected model, otherwise accept it.

Concerning model (3.2) a similar treatment can be performed for each speci®c centered rescaled variable�X �t�=lÿ t=h�=C�t�.


4. Time between failure

The model is thus the following: events of random i.i.d. magnitudes occur at random times T n; n P 1,the waiting times of which form an i.i.d. sequence. An obvious question of interest is the time betweenfailure problem.

How long should one wait before the ®rst event of magnitude greater than a given threshold is observed?This is a standard problem in the context of processes with independent increments (either Gaussian or

L�evy) [6,10,23] or under the more general di�usive Markovian assumptions [23]. Results obtained in thesecontexts have been extended to weakly dependent (ergodic and mixing) processes in [22]. To the authors'knowledge, no such results exist in the context of renewal processes, which are non-Markovian. Extremevalue problems have also been investigated in the context of discrete time chaotic deterministic dynamicalsystems (fully developed chaos and intermittent chaos) in [1]. Other related works in this ®eld are [11] in thecontext of mixing dynamical systems and [3±5] in the context of intermittent expanding dynamical systemsexhibiting a neutral ®xed point.

4.1. Time between failure as an extreme values problem

To address the time between failure problem in the case of a renewal process, we shall compute the ``timebetween failure'', which is de®ned as

T �x� :� inf t�> 0 : XN�t� > x

�; �4:1�

i.e., which is the ®rst time at which some magnitude exceeds the level x > 0. This problem is one of extremevalue theory [9]. To see this, we shall ®rst express the distribution function for the random variablemax�X1; . . . ;XN�t��, where �X ;Xm; m P 1� is the i.i.d. random sequence of amplitudes in a compound re-newal process. We have, for positive x

P max�X1; . . . ;XN�t��

6 x��Xn P 0

P max�X1; . . . ;Xn�� 6 x�P N�t�ÿ � n� �X

n P 1

FX �x�� nP N�t�ÿ � n�

� UN t;� ÿ log FX �x�� 4:2�from the de®nition of function UN �t; k�.

Now, the event ``max�X1; . . . ;XN�t��6 x'' obviously coincides with the event ``T �x� > t'' so that (4.2) givesthe c.p.d.f. of T �x�. Let us now discuss its asymptotics for large times.· h < �1:

The asymptotic distribution of the time to failure is similar to the case of processes with independentincrements belonging to the compound Poisson class. More precisely,

UN �t; k�1=t !t"�1

eÿsN �k�; �4:3�

where the free-energy function sN is de®ned by

/T �ÿsN �k�� exp k; kP 0; �4:4�recalling that the equation /T �p� � exp k is the location of the poles of bUN �p; k�.From the previous expression (4.2), we deduce the following exponential behavior for the time betweenfailure

P�T �x� > t� �t"�1

exp�ÿtsN�ÿ log FX �x��: �4:5�

Examples:(1) The renewal time T is exponentially distributed with mean h, its Laplace transform is/T �p� � 1=�1� hp�.Thus the function sN is sN �k� � �1ÿ exp�ÿk��=h so that sN �ÿ log FX �x�� F X �x�=h. In addition, N�t� is aPoisson process for which P�N�t� � n� � 1

n!exp�ÿt=h��t=h�n and


P�T �x� > t� � UN �t;ÿ log FX �x�� exp�ÿtF X �x�=h�; �4:6�and expression (4.5) is exact (see [16,17] for an application of this point in the context of earthquakesmagnitude data).(2) The renewal time T is Gamma distributed with mean h, its Laplace transform is /T �p� � 1=�1� p�h.Thus the function sN is sN �k� � 1ÿ exp�ÿk=h� so that sN �ÿ log FX �x�� 1ÿ FX �x�1=h . As a result,

P�T �x� > t� �t"�1

expÿt�1ÿ FX �x�1=h� �4:7�

with mean and standard deviation E�T �x�� r�T �x�� 1=�1ÿ FX �x�1=h�.· h � �1:

Here, the asymptotics is of a very di�erent nature. To see what happens, assume, F T �t� � cdtÿd, ast " �1, with d 2 �0; 1�, cd > 0, in such a way that h � �1; then /T �p� � 1ÿ cdpd, as p " 0�. As a result,from the study of (2.16) in the vicinity of p � 0

P�T �x� > t� � UN �t;ÿ log FX �x�� t"�1

cdtÿd

C�1ÿ d�FX �x�F X �x�

; �4:8�

and the time between failure is a power-law with exponent d. The ®rst time at which level x is exceededhas signi®cant probability tail, because events are less and less frequent themselves, making it harder tocross the desired threshold. Note that the expected time between failure ET �x� is itself in®nite in this case.

4.2. Time between failure for extreme events

In some applications, one is interested by the ®rst time by which some amplitude exceeds the level x;where x is itself assumed to be large. Large here means that F X �x� is close to zero, so that crossing thethreshold x is very unlikely. The results (4.5) and (4.8) provide some insight into these questions.· h < �1:

If x is large: sN�ÿ log FX �x�� sN�F X �x�� 1h F X �x�, observing that sN �0� � 0 and s0N �0� � 1=h; we recov-

er the Poisson result (4.6). Therefore the general result is:

1

hF X �x�T �x� !d

x"�1T0 > 0; �4:9�

where T0 is exponentially distributed: P�T0 > t0� � eÿt0 .For example, if X is heavy-tailed with tail exponent a, i.e., F X �x� � caxÿa, ca > 0; a > 0, rescaling thevariable T �x� yields the convergence in distribution

ca

hxÿaT �x� !d

x"�1T0: �4:10�

The time between failure grows like xa for large x; the smaller a is, the heavier the amplitude tails, theshorter the time to overshoot some ®xed large x > 0.

· h � �1:When both x and t tend to in®nity, we have from the identity P�T �x� > t� � UN �t;ÿ log�1ÿ F X �x�� andfrom the study of UN �t; k� in the neighborhood of t � �1, k � 0

P�T �x� > t� ��x;t�"�1

/d�tdF X �x��; �4:11�

where

/d�x� �Xn P 0

1

C�1� nd� �ÿx�n; �4:12�

is the Mittag±Le�er function [14]. De®ning a Mittag±Le�er random variable, say Md, through itsc.p.d.f.


P�Md > t� � /d�td�; �4:13�we get from (4.11) the general result:

F X �x�1=dT �x� !dx"�1

Md > 0: �4:14�

Note that Md is a heavy-tailed as its Laplace transform has the Cole±Cole expression [34]

EeÿqMd � 1=�1� qd�; q P 0 �4:15�for which EeÿqMd �

q"0�1ÿ qd.

A particular case of interest in this article is when F X �x� � caxÿa, ca > 0; a > 0 for large x, i.e., when boththe waiting times and amplitudes are heavy-tailed, but with distinct tail exponents. If such is the case,rescaling the variable T �x� yields the convergence in distribution

�caxÿa�1=dT �x� !dx"�1

Md: �4:16�

The time between failure grows like xa=d for large x, showing again competition between rare and extremeevents.

5. Forward and backward recurrence times: waiting times till the next and since the last event

In this section, we are interested by the random time, say A�t�, separating some ®xed current time t fromthe next event to occur. Also, the time B�t� separating time t from the previous event that occurred will be ofinterest. Times A�t� and B�t� are called, respectively, the forward and backward recurrence times [2,14]. Asnoted in Section 1, we are interested in the behavior for large times t of these random times; these resultswill prove extremely useful in the understanding of the increments process to be discussed in Section 6.These asymptotic behaviors can be derived by exploiting the regenerative structure of the underlyingrenewal process, using identities in distribution and Laplace domain arguments as in Section 2.

5.1. Asymptotic distributions of forward and backward recurrence times

First, note that the forward and backward recurrence times can be expressed in the form

A�t� � T N�t� ÿ t and B�t� � t ÿ T N�t�ÿ1: �5:1�Note also that the forward and backward recurrence times A�t� and B�t� can be self-generated through thefollowing identities in distribution:

A�t� �d �T ÿ t� � 1�T > t� � A�t ÿ T � � 1�T 6 t�; �5:2�B�t� �d t � 1�T > t� � B�t ÿ T � � 1�T 6 t�: �5:3�

The variable S�t� :� A�t� � B�t� is the width of the waiting time interval which contains t. Let now, withq :� �q1; q2�

U�t; q� :� Eeÿ�q1A�t��q2B�t�� 5:4�stand for the Laplace transform of the joint distribution of �A�t�;B�t�� From (5.2) and (5.3), the Laplacetransform of U��; q� is

U�p; q� � 1

p � q2 ÿ q1

/T �q1� ÿ /T �p � q2�1ÿ /T �p�

: �5:5�

Again, two cases are to be distinguished.


· h < �1:Here, /T �p� � 1ÿ hp, as p " 0�. Hence,

bU�p; q� �p"0�

1

h/T �q1� ÿ /T �q2�

q2 ÿ q1

:� 1

h/T �q1; q2�: �5:6�

This means that �A�t�;B�t�� converges in distribution, as t " �1, to a random vector, say �A;B�, whoseLaplace transform is 1

h /T �q1; q2�. It may easily be shown that the corresponding density of �A;B� at point�a; b�, say g�a; b�, expresses in the symmetric form

g�a; b� � 1

hfT �a� b�: �5:7�

To summarize, for recurrent positive processes

A�t�;B�t�� !dt"�1�A;B� with density g�a; b� � 1

hfT �a� b�: �5:8�

Many interesting consequences may be derived from this. First the following holds on the marginaldistributions

A�t� !dt"�1

A with d:f : : g�a� � 1

hF T �a�; �5:9�

and

B�t� !dt"�1

B with d:f : : g�b� � 1

hF T �b�: �5:10�

These are the main asymptotic results under the hypothesis h < �1. We supply a few additional in-formations worth being underlined.Additional results and properties in the case h < �1(1) It may also be easily checked that hA :� EA < �1 if and only if time T has ®nite variance:r2�T � < �1. In more precise terms, hA is easily found to be in this case

hA � 1

2h�r2�T � � h2�: �5:11�

(2) Concerning the conditional variables A�t� given B�t� � b and B�t� given A�t� � a, they converge indistribution to random variables with respective density

gB�b�a� � fT �a� b�F T �b�

and gA�a�b� � fT �a� b�F T �a�

: �5:12�

Thus the forward (backward) recurrence time given the backward (forward) recurrence time is ®xed hasknown asymptotic density for large current time t, if the equilibrium is reached. The symmetry observedin (5.8) is broken in the process of conditioning.(3) In the critical case where the renewal time T is exponentially distributed, a stronger result actuallyholds:the distribution of the joint vector �A�t�;B�t�� is g�a; b� � 1

h fT �a� b� � 1

h2 expÿ�a� b�=h for all times t.The marginal densities of A�t� and B�t� are exponential: g�a� � 1=h expÿa=h and g�b� � 1=h expÿb=hfor all times t and the conditional densities are gB�b�a� � 1=h expÿa=h and gA�a�b� � 1=h expÿb=h, in-dependent of the conditioning: the backward and forward recurrence times are independent as a result ofthe ``memory-less'' character of the exponential distribution, which may be expressed byPT>t�T 6 t � s� � P�T 6 s�, upon conditioning by the event T > t:(4) Note also the following symmetry-breaking property which has been underlined in [29] in the contextof earthquake magnitude data in response to the question Q:


``the longer it has been since the last earthquake, the longer the expected time till the next''

If time T has super-exponential tails (such as absolute Gaussian, Rayleigh or Weibull with exponentgreater than one distributions), i.e., tails thinner than the exponential distribution, the answer to Qisnegative; on the contrary, the answer to Q is positive for sub-exponential distributions (such as log-normal, Weibull with exponent smaller than one distributions or power-law with exponent greater thanone), i.e., with tails fatter than the exponential distribution (but not too fat so as to insure ET < �1):the exponential tail fall-o� serves as a crossover with respect to question Q. If T is exponential, thebackward and forward recurrence times are independent.(5) From (5.8) the joint distribution of �S :� A� B;D :� Aÿ B� is found to be

g�s; d� � 1

2hfT �s�1�s > jdj� �5:13�

at point �s; d�.Thus, the limit distribution of the sum S�t� :� A�t� � B�t� has density

g�s� � 1

hsfT �s�1�s > 0� �5:14�

with mean value 2h (the waiting time paradox).The limit distribution of the di�erence D�t� :� A�t� ÿ B�t� has the symmetric density at D � d 2 R

g�d� � 1

2hF T �jdj� �5:15�

which corresponds to the real-valued Fourier transform

1

2i-h/T �� ÿ i-� ÿ /T �i-��: �5:16�

(6) Finally, the variable R :� A=B which is the ratio between forward and backward asymptotic recur-rence times has the universal Pareto c.p.d.f. (with tail exponent one) at R � r 2 R�

P�R > r� � �1� r�ÿ1; �5:17�

independently of the particular form of the density fT of time T.

· h � �1:In this case, i.e., for recurrent null processes, �A�t�;B�t�� does not converge in distribution, as t " �1.Under our rare events hypothesis (2.6), we get

bU�p; q� �p"0�

1

cdpH

/T �q1� ÿ /T �p � q2��p � q2 ÿ q1� : �5:18�

As it may easily be checked, in the large t limit, the corresponding density of �A�t�;B�t�� at �a; b�, sayg�t; a; b�, expresses under the asymmetric form

g�t; a; b� � 1

cdC�d�C�1ÿ d� fT �a� b��t ÿ b�dÿ1 �5:19�

extending (5.8) to d < 1. Here, the support of g�t; a; b� is �a > 0� � �06 b < t� as B�t� cannot exceed t.Proceeding now to scaling in time yields for the recurrent null process for which fT �t� �

t"�1cddtÿ�d�1�; we

get

1

tA�t�;B�t�� !d

t"�1A0;B0� �; �5:20�


where the vector �A0;B0� admits the probability density on R� � �0; 1�

g0�a0; b0� � dC�d�C�1ÿ d� �a0 � b0�ÿ�d�1��1ÿ b0�dÿ1

: �5:21�

As a consequence, the following results are easily shown to hold for the marginal distributions of A�t�and B�t�

1

tA�t� !d

t"�1A0 with d:f : : g0�a0� � 1

C�d�C�1ÿ d� aÿd0 �1� a0�ÿ1; �5:22�

and

1

tB�t� !d

t"�1B0 with d:f : : g0�b0� � 1

C�d�C�1ÿ d� bÿd0 �1ÿ b0�dÿ1 �5:23�

with �a0 > 0� and �b0 2 �0; 1��.This last density is identi®ed with the generalized arcsine law, which appears here and there in ¯uctuationtheory [7,21].

Additional results and properties in the case h � �1We underline some important properties of the model in the situation under concern.In sharp contrast with the case h < �1, the backward and forward recurrence times diverge jointly andmarginally: only a rescaled version of these variables converge. In addition, owing to the rareness hy-pothesis, some asymmetry is found in the limit distributions.(1) From (5.22), g0�a0� � aÿ�d�1�

0 for large a0. With d < 1, we get EA0 � �1.(2) From (5.19), the conditional variable A�t� given B�t� � b still converges in distribution to a randomvariable with density

gB�b�a� � fT �a� b�F T �b�

�5:24�

just like in the positive recurrent case. Thus the forward recurrence time given the backward recurrencetime is ®xed has known asymptotic density for large current time t. Nevertheless, in this case, theconditional average of A given B � b diverges and the answer to question Q is ill-de®ned; rather, theconditional median value should be considered instead.To take an example, assume F T �a� � �1� a�ÿd

, i.e., a Pareto model with exponent d 2 �0; 1� for T. Then,from (5.24), we get the conditional p.d.f.

PB�b�A6 a� � �F T �b� ÿ F T �a� b��=F T �b� � 1ÿ �1� a1� b

�ÿd

and the conditional median value of A given B � b is: m � �21=d ÿ 1��1� b�, which increases with b andassertion Q should be replaced by ``the longer it has been since the last earthquake, the longer the mediantime till the next''.

5.2. Delayed renewal processes and stationarity

In this sub-section, we investigate the important question of stationarity of renewal processes, under thehypothesis that some initial time delay is present. We shall again supply a di�erent study depending onh < �1 or h � �1.

· h < �1:Consider now a delayed process for which the origin of time is not an instant at which some event oc-curred; rather, one has to wait a random time T0, before the ®rst occurrence of an event is seen and beforethe mechanics of a pure renewal process proceeds.


Time T0 is assumed independent of the i.i.d. waiting times �T ; Tm; m P 1�, with density fT0.

Under this hypothesis, the distribution of the delayed process, say Xd�t�, may be obtained through

Xd�t� �d 0 � 1�T0 > t� � X �t ÿ T0� � 1�T06 t�

in terms of the shifted underlying pure process X �t�.In a similar way, the distribution of the forward recurrence time Ad�t� of this delayed process X

d�t� maybe expressed as

Ad�t� �d �T0 ÿ t� � 1�T0 > t� � A�t ÿ T0� � 1�T06 t� �5:25�

in terms of the shifted forward recurrence time A�t� of the underlying pure renewal process. From thisobservation, it is easy to derive the following important fact:If the distribution of T0 is the one of the asymptotic forward recurrence time A, then Ad�t��d A, for alltimes t > 0. In other words, the distribution of the forward recurrence for the delayed process, withT0�d A, is invariant with time. Such delayed renewal processes are often called stationary, for reasonsto appear later.We shall brie¯y indicate how this works:

Let

Ud�t; q� :� EeÿqAd �t� �5:26�

stand for the Laplace transform of the distribution of Ad�t�. From (5.5) and (5.25), the Laplace transformof Ud��; q� is

Ud�p; q� � 1

qÿ p/T0�p�

�ÿ /T0

�q� � /T0�p�

1ÿ /T �p��/T �p� ÿ /T �q��

�; �5:27�

where /T0�p�; /T �p� are the Laplace transforms of the distributions of T0 and T.

If T0�d A, then, from (5.9), /T0�p� � 1

ph �1ÿ /T �p�� EeÿpA. Inserting this expression in (5.27) yields aftersome easy algebraic work

Ud�p; q� � 1

pqh�1ÿ /T �q��: �5:28�

This shows that Ud�t; q� � 1qh �1ÿ /T �q�� for all times t > 0; so that Ad�t��d A for all times t > 0.

Thus, the distribution of the forward recurrence time of the delayed process for which T0�d A is invariantwith time.Let us now brie¯y investigate if this can be possible in the case h � �1.

· h � �1:Here, A�t� does not converge. However, A0�t� :� 1

t A�t� converges in distribution to a random variable A0

whose distribution is given in (5.22). Now the question, extending the one in the h < �1 case, is as fol-lows:Is it possible to choose the distribution of T0 in such a way that Ad

0�t� :� 1t Ad�t��d A0, for all times t > 0 ,

leaving invariant the distribution of the rescaled forward recurrence time Ad0�t� of the delayed process?

Setting q2 � 0, q1 � q in (5.5) yields the marginal Laplace transform for A�t�:

bU�p; q� � 1

qÿ p/T �p� ÿ /T �q�

1ÿ /T �p�: �5:29�

Under the rare event hypothesis (2.6)

U�p; q� ��p;q�"0�

� 1

qÿ pqd ÿ pd

pd: �5:30�


Now, if the announced property is to be satis®ed, there must be a choice of T0 expressed in terms of T forwhich

Ud�p; q� � 1

qÿ pqd ÿ pd

pd�5:31�

for any choice distribution of T. Indeed, if this were so, for all times t > 0

EeÿqAd0�t� � Ud�t; q=t� � EeÿqA0 �5:32�

Thus, were (5.31) to be ful®lled, Ad0�t� :� 1

t Ad�t��d A0 for all times t > 0, which is the required property.However, from the expression (5.27), it is easy to be convinced of the fact that there is no choice of /T0

�p�as a function of /T �p� for which (5.31) is satis®ed for any /T �p� bound to satisfy /T �p� �

p"0�1ÿ cdpd. Thus,

if h � �1, i.e., for null recurrent processes, the answer to our question is negative, in sharp contrast withthe previous case h < �1.

6. The increments process

Let us begin with some introductory remarks. Poisson processes are not stationary processes; however,they have stationary increments. The renewal processes' generalization of Poisson processes has a drasticimpact on the stationary character of the increments process, which is de®ned as

X �t; t � s� :� X �t � s� ÿ X �t� �6:1�

with s > 0; t > 0. Here s > 0 is the length of the interval over which the increment has to be evaluated. Theunderstanding of the increments process is important for the following reason, among others: assume theprocess under interest is observed starting from time t > 0 and onwards, over a span of time s > 0; there isno data available between time zero (assumed to be known) and switching time t. What conclusions maynow be drawn from these observations, especially if the starting observational time t is large?

The answer to this question deserves a particular study of the increments process which we now supply.The distribution of the increments process may be generated in the following self-understanding way,

using the forward recurrence time A�t�:

X �t; t � s� �d 0 � 1�A�t� > s� � �X � X �t � A�t�; t � s�� 1�A�t�6 s�: �6:2�

This identity states that the increment between time t and t � s is zero if the forward recurrence time from texceeds s, whereas, otherwise, it is the independent sum of a magnitude which occurred at time t � A�t� plusa statistical copy of the increment process between times t � A�t� and t � s. From this expression, oneimmediately realizes that, except for very special cases (T exponentially distributed and X �� compoundPoisson), the distribution of X �t; t � s� will depend on both s and t, and not on s only: the incrementsprocess is not stationary (except possibly in some asymptotic sense). Let us again distinguish between thetwo cases h < �1 and h � �1.

· h :� ET < �1:It has been shown that A�t� !d

t"�1A with density d.f.: g�a� � 1

h F T �a�. If this asymptotic regime is reached,

we shall see that it is possible to supply a detailed statistical study of the increment in the large elapsedtime t limit

X �s� :� limt"�1

X �t; t � s�; �6:3�

as a function of the length s > 0 of the interval over which it has to be evaluated. Very di�erent as-ymptotic behaviors are observed, depending now on whether or not the variance r2�T � is ®nite. Theseconditions are equivalent to the existence or not of a mean value for the limit forward recurrence time A


given by (5.11). The increments process of a renewal process is not stationary in the sense thatX �t; t � s��d X �0; 0� s� does not hold in general.The only cases where the increments are strictly stationary is when A�t��d A for all times, i.e.,(1) when T is exponentially distributed if the renewal process is pure (or delayed); in this case, A is itselfexponentially distributed as was shown in Subsection 5.1.(2) When the process is delayed and stationary, i.e., when the distribution of the time delay T0 coincideswith the one of the limit forward recurrence time A of the underlying pure renewal process: T0�d A. Ifsuch is the case: Ad�t��d A for all times t > 0 and the increments process X d�t; t � s� :� X

d�t � s�ÿX

d�t� is independent of time. In other words, for any t > 0

Xd�s� :� X d�0; 0� s� �d X d�t; t � s� �d X d�s� :� lim

t"�1X d�t; t � s�:

· h :� ET � �1:It has been shown that 1

t A�t� !dt"�1

A0 with density d.f. g0�a0�: the forward recurrence time grows like time t,

as a result of the rareness hypothesis of the events, so that the larger the time t elapsed from the origin oftime, the larger the forward recurrence time from t. This forces us to study the increments of the processon intervals s whose length is proportional to the elapsed time t from the origin: s � ts0, s0 > 0, in order

to capture enough events. If the asymptotic regime 1t A�t� !d

t"�1A0 is reached, we shall supply a simpli®ed

statistical study of the asymptotic increment for large elapsed times t

X0�s0� :� limt"�1

X �t; t�1� s0��; �6:4�

as a function of the relative length s0 > 0.If instead the process is delayed, we saw that there was no way to choose the distribution of T0 in such away that Ad

0�t��d A0 for all times, i.e., delayed null recurrent process cannot be stationary. Thus there isno particular interest for delayed process in this case.

Let us enter into computational details. Let

W�t; t � s; k� :� E expÿkX �t; t � s� �6:5�stand for the Laplace transform of the increments process. Using (6.2), one deduces the following integralequation

W�t; t � s; k� � P�A�t� > s� � /X �k�Z s

0

W�t � a; t � s; k�g�t; a� da; �6:6�

where g�t; a� is the density of A�t� at A�t� � a. Taking the Laplace transform with respect to s:

W�t; q; k� :�Z �1

0

eÿqsW�t; t � s; k� ds; �6:7�

we get the renewal equation

W�t; q; k� � 1

q�1ÿ /A�t; q�� /X �k�

Z �1

0

eÿqaW�t � a; q; k�g�t; a� da; �6:8�

where /A�t; q� :� EeÿqA�t� is the Laplace transform of A�t�, see (5.4). We shall now distinguish between thetwo cases h < �1 and h � �1.

6.1. Case h :� ET < �1

For large times t for which g�t; a� � g�a�, letting W�s; k� :� E expÿkX �s�, with X �s� de®ned by (6.3) andW�q; k� the Laplace transform of W��; k�, Eq. (6.8) reduces to

W�q; k� � 1

q�1ÿ /A�q�� /X �k�

Z �1

0

eÿqaW�q; k�g�a� da: �6:9�


Hence, if, we get

W�q; k� � �1ÿ /A�q��q�1ÿ /X �k�/A�q��

; �6:10�

where /A�q� is the Laplace transform of the limit forward recurrence time A, i.e., /A�q� :� EeÿqA ��1ÿ /T �q��=�hq�, in terms of the Laplace transform /T �q� of the waiting time itself.

The latter formula is strongly reminiscent of the one (2.16) with /A replacing /T . Following Section 3,di�erent asymptotic behavior of X �s� (as a function now of the length s > 0) are to be expected dependingon hA :� EA < �1 or hA � �1. It turns out that these conditions are equivalent to r2�T � < �1 orr2�T � � �1, in terms of the variance of the waiting times. Let us discuss this point.

6.1.1. Case hA < �1 �or r2�T � < �1�Here,

W�s; k� �s"�1

expÿssX �k�; �6:11�

where the free energy function sX �k� is now implicitly de®ned through /A by

/A�ÿsX �k�� /X �k�ÿ1: �6:12�

Again, we shall need to distinguish two sub-cases, depending on the mean value l :� EX of the localmagnitudes:· l < �1: /X �k� �

k"0�1ÿ lk, and sX behaves like

sX �k� �k"0�

lhA

k �6:13�

Following (3.2), we get

1

s=hA

X �s�l!Pr

1:s �6:14�

· l � �1.X is assumed here to be heavy-tailed with power-law exponent a less than one. The function sX �k� nowbehaves like

sX �k� �k"0�

ca

hAka: �6:15�

From (3.8), we get the convergence to a L�evy distribution

1

�jas=hA�1=aX �s� !d

s"�1La: �6:16�

6.1.2. Case hA � �1 (or r2�T � � �1)Here, the variance of the time between consecutive events is assumed in®nite. In this case, there is no pole

in W�q; k� compatible with the requirement that kP 0 and the asymptotics is of a very di�erent nature.We shall now assume heavy-tailedness of the forward recurrence time A, i.e., P�A > a� �

a"�1cdA aÿdA for

some tail exponent dA 2 �0; 1�.In terms of T, this means P�T > t� �

a"�1cdA tÿ�1�dA� so that the waiting times have ®nite mean value but

in®nite variance.Proceeding as in (3.13) and (3.19), we get, again distinguishing between two sub-cases

· l < �1 :

jdA

sdA

X �s�l!d

s"�1IdA : �6:17�


· l � �1 :

1

jdA;asdA=aX �s� !d

s"�1LdA;a; �6:18�

where the variables �IdA ; LdA;a� and constants �jdA ; jdA;a� are de®ned in (3.13), (3.19) and (3.23), substi-tuting dA to d.

6.1.3. Partial conclusionsIf only the increments of a pure and positive recurrent renewal process are available over some lapse of

time s, and if the starting observational time t is ``far enough'' from the origin of times (t� t0) so that theequilibrium of A�t� may be considered as achieved, various large observational time s behavior are pre-dicted for this variable, depending this time on the ®niteness (or lack thereof) of the variance of the waitingtimes. Note that in order to apply these results to real data, one needs to control the speed of convergenceof A�t� to A in order to be able to give a quantitative meaning to the phrase for large t.

Following Subsection 3.5, inspection of the empirical log±log plots of both the variation of the numberand amplitudes of events over large observational times s from t may be used to decide whether thisvariance is ®nite or in®nite (dA < 1 or not) and if the events' amplitude is or is not heavy-tailed with ex-ponent a < 1.

In situations for which dA < 1 and/or a < 1, the empirical log±log plots of both n�s� :�limt"�1 n�t � s� ÿ n�t� and x�s� :� limt"�1 x�t � s� ÿ x�t� against observational time s may be used to derivean estimator of both tail exponents dA and a, simply by inspection of their slopes for large s.

If in addition the process under study is delayed and stationary the same conclusions hold for any t > 0at which the observation procedure has been started.

6.2. Case h � �1

Setting s � ts0 in (6.6), we get

W�t; t�1� s0�; k� � P1

tA�t�

�> s0

�� /X �k�

Z ts0

0

W�t � a; t�1� s0�; k�g�t; a� da: �6:19�

Performing the change of variable a � ta0 in the integral of the right-hand side, we get

W�t; t�1� s0�; k� � P1

tA�t�

�> s0

�� /X �k�

Z s0

0

W�t�1� a0�; t�1� s0�; k�tg�t; ta0� da0; �6:20�

where tg�t; ta0� is the density of the variable �1=t�A�t�; which converges in distribution to the randomvariable A0. Hence, if W0�s0; k� :� E expÿkX0�s0�, with X0�s0� de®ned in (6.4) and if W0�q0; k� is the Laplacetransform of W0��; k�, we get for large elapsed times t

W0�q0; k� � 1

q0

1ÿ /A0�q0�

1ÿ /X �k�/A0�q0� ; �6:21�

where /A0�q� is the Laplace transform of the limit rescaled forward recurrence time A0, i.e.,

/A0�q0� � Eeÿq0A0 .This equation is strongly reminiscent of the ones, (2.16)±(6.10) with /A0

replacing /T and /A, respec-tively.

Recalling that EA0 � �1 as a result of P�A0 > a0� �a0"�1

aÿd0 , we get from ((3.13)±(3.19)), distinguishing

only between two the sub-cases l < �1 and l � �1:

· l < �1:

jd

sd0

X0�s0�l

!ds0"�1

Id: �6:22�


· l � �1:

1

jd;asd=a0

X0�s0� !ds0"�1

Ld;a: �6:23�

6.2.1. Partial conclusionsIf only the increments of a pure and null recurrent renewal process are available over some span of time

s, and if the starting observational time t is far enough from the origin of times so that the equilibrium ofA0�t� :� �1=t�A�t� may be considered as achieved, various large relative observational time s0 behavior areavailable. Note that this asymptotic regime may require prohibitive amounts of data, since both t, s and theratio s=t need to be large. Here too, a more detailed study of the speed of convergence of A0�t� to A0 isneeded.

7. Discussion of main results

Let us summarize the main results of the preceding sections for CTRWs:1. In the context of power-law type distributions for both amplitude and holding time, suitably rescaled

magnitude partial sums are shown to converge to fractal-type distributions, such as L�evy, inverse L�evyand other related laws. Whereas the heavy-tailed amplitudes hypothesis tends to accelerate the partialsums growth with time, heavy-tailed waiting times tend to slow this process, with a competition betweenthe two. A comparative study with the standard case where both distributions have ®nite mean is pro-vided. The statistics associated with the di�erent limit laws can be used to decide between alternative tailassumptions in the face of empirical data. However, these results are predicated on the hypothesis thatthe partial sum is observed starting from the origin of time.

2. Asymptotic results in both space and time directions are provided for the time between failure problem,which is the ®rst time at which some magnitude exceeds a given (possibly high) threshold.

3. Using asymptotic results on backward and forward recurrence times, the physically realistic situationwhen the process is only observed starting from time t > 0 over a span of time s > 0 is addressed. Ifthe expected holding time is ®nite, and for large starting observational time t , the increments processon intervals of length s exhibit di�erent large s asymptotic regimes depending on whether the varianceof the waiting time is ®nite or in®nite. Under the same conditions for t, if the expected holding time isnow in®nite, limit distributions for suitably rescaled increments will be attained only on the conditionthat both t, s and the ratio s=t are large.

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