Mandelbrots 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence

Mandelbrot’s 1 /f fractional renew al mo dels of 1963-67: The non-ergo dic missing link b et w een c hange p oin ts and long range dep endence. Nic ho las Wynn W atkins 1 2 3 4 5 1 Cen t re for the Analysis of Time Series, London School of Economics, London, UK, N.Watkins2 @lse.ac.uk , 2 F a cu lty of Mathematics, Computing and T ec h nology , Op en Universi ty , Milton Keynes, UK, 3 Cen t re for F usion, Space an d Astrophysics, Un iversit y of W arwic k,UK 4 Universit¨ at P otsdam, Institut f¨ ur Ph y sik und Astronomie, Campus Golm, P otsdam-Golm, Germany 5 Max Planc k Institute for the Physics of Complex S y stems, Dresden, German y . Abstract. The problem of 1/f noise has b een with us for about a cen- tury . Because it is so often framed in F ourier spectral language, the most famous solutions hav e tended to b e the stationary long range dep endent (LRD) mod els such as Mandelbrot’s fractional Gaussian noise. In v iew of the increasing importance to ph y sics of non - ergodic fractional renew al mod els, I present preliminary results of my research into the history of Mandelbrot’s very little known work in t hat area from 1963-67. I sp ec- ulate ab out how the lack of aw areness of this wo rk in the physics and statistics communities ma y hav e affected the developmen t of comp lexity science; and I d iscuss the differences betw een the Hurst effect, 1/f n oise and LRD, concept s which are often treated as equiv alent. Keywords: Long range dep endence, Mandelbrot, change p oin ts, frac- tional renew al mo dels, weak ergo d icit y breaking 1 Ergo dic and non-ergo dic solutions to the parado x of 1/f noise “The pr oblem of 1 /f noise” has b een with us for ab out 100 years since the pio- neering work of Schottky a nd Johnson [10,9,3 0]. It is usually framed a s a spectral paradox, i.e.“ how can the F our ier sp ectral densit y S ′ ( f ) of a stationa ry proc ess take the form S ′ ( f ) ∼ 1 /f and thus be singular at the origin (or equiv a len tly how can the auto correla tion function “blow up” at lar ge lag s and thus not b e summable) ? ”. When a pr oblem is s een this way , the so lution will a ls o tend to be sought in sp ectral terms. The desir e o f for a solution to the problem with a satisfying level of generality increa s ed in the 19 50s with the reco gnition of an analogo us time domain effect (the Hurst phenomenon) seen in the sta tis tica l growth of range in Nile minima [9]. The first statio na ry solution whic h could ex- hibit the Hurst effect a nd 1 /f no ise was pr esen ted b y Mandelbr o t in 1965 using 2 fractional Gaussia n noise (fGn), the incr emen ts o f fractional Br o wnian motion (fBm), and was subsequently developed with V an Ness and W allis , particularly in the hydrological context [9,19]. fGn is a s tationary erg odic pro cess, for whic h a pow er spectr um is a natural and well-defined co ncept, the par ado x here re s ides in its singular b ehaviour a t zero. How ever, in the la st t wo decades it has incr easingly been realis e d in physics that another class of models , the fra c tional renewal pro cesses , can also give 1 /f noise in a very different wa y[21]. Physical in ter est has come from phenomena such a s weak ergo dicit y brea king (in e.g. blinking quantum do ts [29,25,28]) and the related question of how many differen t classe s of mo del can share the c ommon prop erty of the 1/ f spectra l shape (e.g. [2 6,27]). In view of this resurgence of ac- tivit y , m y first aim in this pap er is to repor t (Sectio n 2) preliminar y results fro m m y historica l research which has found, to my gr eat surpris e , that the dichotom y betw e e n er godic and non-er go dic origins for 1 /f sp ectra w a s not o nly recognised but als o published a bout by Mandelbrot 5 0 years ago in some still remark ably little known work [4,1 5,16,17]. He ca rried it out in parallel with his seminal work on the e r go dic , s ta tionary fGn mo del. In these pap ers, and the br idg ing es s a ys he wrote when he revisited them late in life for his collected Sele cta volumes, particularly [1 8,19], he develop ed and published a s eries of fractional renewal mo dels. In these the p erio dogram, the empirical auto correlatio n function (acf ), and the obser v ed waiting time distributions, all g ro w in extent with the length of time over whic h they are measure d. He explicitly [1 7] dr ew attention to this non-ergo dicity a nd its orig ins in what he called “conditional” stationarity . He explicitly contrasted the fractional renewal mo dels with the stationar y , ergo dic fGn which is to da y very muc h b etter known to physicists, geosc ien tists and many other time series ana lysts [3 ,9]. Mandelbrot’s work at IBM was itself in para llel with other developments, one notable example b e ing the work o f Pierr e Mertz [22,23] at RAND on mo delling telephone erro r s, so m y preliminary rep ort is not an attempt to a ssign prior it y . I hop e to return to the histor y of this perio d in more detail in future a rticles. My next purpos e (Section 3) is to clarify the subtle differences b etw een 3 phenomena: the empirical Hurst effect, the a ppeara nce of 1 / f no is e in per i- o dograms, and the co nc e pt of LRD as embo died in the stationa ry ergo dic fGn mo del, and to set o ut their hier arch y with resp ect to each o ther, aided in part by this histo rical pers pective. This pap er will not dea l with another p ossibility- m ultiplica tiv e mo dels-[18,26], though I do o f co urse reco gnise that they r emain a very impo rtan t alternative source of 1 /f b e haviour, par ticularly that arising from turbulent c a scades. I will a lso not be co nsidering 1 /f - type sp ectra ar ising from nonstationa ry self-similar walks suc h a s fractional Brownian motion. I will (Section 4 ) conclude b y specula ting on the how the relative neg lect of[4,15,16,17] at the time of their publica tio n may have had lo ng-term effects. 3 1.1 fGn and the fractional renewal pro cess compared fGn [3] is effectively a deriv ative of fractional Brownian mo tio n Y H, 2 ( t ): Y H, 2 ( t ) = 1 C H, 2 Z R dL 2 ( s ) K H, 2 ( t − s ) (1) which in turn extends the Wiener pro cess to include a self-similar , memory kernel K H, 2 ( t − s ), where K H, 2 ( t − s ) = [( t − s ) H − 1 / 2 + − ( − s ) H − 1 / 2 ] (2) th us g iving a decaying, non-zero weight to all of the v alues in the time integral ov er dL . In consequence fGn shows long range dep e ndence, and has indeed beco me a very imp ortant pa radigmatic mo del for LRD. The attention paid to its 1 /f sp ectrum, a nd long-ta iled acf, a s diagno stics of LRD, has often led to it b eing forgotten that its stationarity is an eq ually essential ingredient for LRD in this sense. Intuitiv ely one can see that without stationar it y there can b e no LRD bec ause there is no infinitely long pas t histor y over which the pr ocess can be depe ndent. Mo dels like fGn, a nd a lso fr actionally integrated noise (FIN ) a nd the autoregr essive fractionally in tegra ted moving av e r age (ARFIMA) pro cess, which hav e b een widely studied in the statistics co mm unity (e.g. [2,3]) exhibit LRD by c onstruction , i.e. stationarity is assumed a t the outset in defining them. While undenia bly impo rtan t to time s eries ana ly sis and the development of complexity science, we can alrea dy see from the re striction to stationary pro- cesses that the LRD c oncept, at least a s embo died in fGn, will b e insufficien t to describ e the who le ra nge of either 1 /f or Hurst b ehaviour that obser v ations may pr esen t us with. F ull awareness of this limitation has b een slow becaus e of three wides pread, deeply-ing rained, but unfortunately erroneo us beliefs: i) that an o bserved F ourie r per iodog ram can always be taken to es timate a p ow er sp ec- trum, ii) that the F ourier transform of an empirica lly obtained per iodog ram is always a meaningful es timator of an auto correlatio n function, a nd iii) that the observ ation of a 1/f F ourie r per iodog ram in a time series must imply the kind of long range dep endence that is em b o died in the ergo dic fractional Gaussian noise mo del. The first tw o b eliefs ar e routinely cautio ned aga inst in any go o d cours e or bo ok on time ser ies analy s is, including classics like Be nda t’s [1]. The third belie f remains highly to pical, how ever, b e cause it is only relatively r ecen tly being appreciated in the theoretica l physics litera tur e just how distinct tw o paradig - matic classes o f 1/f noise mo del ar e, and ho w these differences rela te not only to LRD but also to the fundamen tal physical que s tion of weak ergo dicit y breaking (e.g. [6,21,25]). The second pa radigm for 1/f noise mentioned above is the frac tional renew a l class, which is a descendent of the class ic r andom telegraph mo del [1], a nd so lo oks at first sight to be sta tionary and Marko via n, but has switching times at p ow er law distributed int e r v als . A particula r ly w ell studied v ar ian t is the alternating fractal renewal pro cess (AFRP , e.g. [13,14]), which is als o closely 4 connected to the r enew al r ew ar d pro cess in mathematics. When studied in the telecommunications co n text, how ever, the AFRP has often had a cutoff applied to its switching time distribution for large times to allow analytical tractability . The use of a n upper cutoff unfortunately masks s o me of its most physically int er esting b eha vio ur, b ecause when the cutoffs are not us ed the pe r iodog ram, the e mpir ical acf, and observe d w a iting time distributions, a ll gr ow with the length of time over which they ar e measur ed, rendering the pro cess b oth non- ergo dic and non-s tationary in an impor tan t s ense (Mandelbrot preferr ed his own term “ conditionally stationary” ). In particula r, Ma ndelbrot stressed that the pr o cess no lo nger ob eys the neces sary conditions o n the Wiener-Khinchine theorem for its empirical pe riodo gram to be interpreted as an estimate o f the power sp ectrum. This prop ert y of weak e r go dicit y br e aking (named by B o uc haud in the ea rly 1990 s [6]) is now attracting muc h interest in physics, see e.g. Niema nn et al [25], on the res o lution of the low frequency cutoff paradox, and subs equen t developmen ts [12,7,26,2 7]. The existence of this alter nativ e, nonsta tionary , nonergo dic fractional re- newal mo del mak es it clear that there is a difference betw een the obser v ation of an empirical 1/ f noise a lone, and the pr esence o f the type of LRD that is em- bo died in the stationa ry ergo dic fGn mo del. W e will develop this p oint further in section 3, but will first g o back to the 1960 s and Mandelbrot’s twin tr ac ks to 1 /f . 2 Mandelbrot’s fractional renew al route to “1/f” What seems to have go ne almost completely unnoticed, is the re ma rk a ble fact that Mandelbrot was not only aw are of the distinction betw een fGn and frac- tional re new al mo dels [18,19], but also published a nonstationa ry mo del o f the AFRP type in 196 5 [1 5,16] a nd had explicitly discussed the time depe ndence of its p ow er sp ectrum as a symptom o n non-ergo dicity by 1 967 [17]. There are 4 key pap ers in Mandelbro t’s consider ation of fractional r enew al mo dels. The fir s t, cowritten with ph ysicist Ja y B erger [4], app eared in IBM J our- nal o f Resear c h and Developmen t. It dealt with er rors in telephone circuits, and its k ey point p oint was the power law distribution of times b et ween er r ors, which were themse lves ass umed to take discrete v alues. Switching models w er e alr eady being lo oked at, a nd the author s ackno wledged that Pierr e Mertz of RAND had already studied a p ow er la w switching mo del [22], but Mandelbrot’s early exp o- sure to the e x tended central limit theorem, a nd the fact that he was studying heavy tailed mo dels in econo mics and neuroscienc e a mong other a pplica tions, evidently help ed him to s e e their br o ader significance. The second, [15] was in the IEEE T ra nsactions on Communication T ech no l- ogy , and essentially also used the model of Berge r and Mandelbrot. The abstract makes it clear that it describ es: ... a mo del of c ertain r andom p erturb ations that app e ar to c ome in clus- ters, or bur s t s. This wil l b e achieve d by intr o ducing the c onc ept of “self- similar sto chastic p oint pr o c ess in c ontinuous time.” The r esulting me ch- 5 anism pr esent s fascinating p e culiarities fr om the mathematic al viewp oint. In or der to make t hem mor e p alatable as wel l as to help in the se ar ch for further develop m ents, the b asic c onc ept of “c onditional st ationari t y” wil l b e discusse d in gr e ater detail than would b e strictly ne c essary fr om the viewp oint of the imme diate engine ering pr oblem of err ors of tr ans- mission. It is clear that by 196 5 Ma ndelbrot had co me to a ppreciate that the appli- cation o f the F o urier per iodog ram to the fractional r e new al pro cess would give ambiguous r esults, saying in [15] that: The now classic al te chnique of sp e ctr al analysis is inapplic able t o the pr o c esses examine d in this p ap er but it is sometimes unavoidable that otherwise exc el lent sp e ctr al estimates b e appli e d in this c ontex t. Another public ation of the aut hor[ t hat p ap er’s R ef 18] is devote d to an examina- tion of the exp e cte d outc omes of such op er ations. This wil l le ad to fr esh c onc epts that app e ar most pr omising inde e d in the c ont ext of a statistic al study of turbulenc e, ex c ess noise, and other phenomena when inter esting events ar e intermittent and bunche d to gether (se e also [that p ap er’s R ef 19]). The thir d key pap er, the “other publication ... Ref 1 8 ”, resulted from an IEE E conference talk in 1965. It [1 6] is now av ailable but in the p ost ho c edited form in which all his pape rs app eared in his Selecta[18,19]. “Reference 19”, mea n while, seems origina lly to have b een intended to b e a pa per in the physics litera ture, the fate of which is not clear to me but who se role was e ffectiv ely taken ov er by the fourth key pap er [17]. With the proviso that the Se le cta version of [16] may not fully reflect the orig inal’s co n tent, one can no netheless see that b y mid- 1965 Mandelbrot w a s alrea dy fo cusing o n the implications for erg odicity of the conditional statio narit y idea. He remar k ed that: In other wor ds, the existenc e of f θ − 2 noises chal lenges t he m athemati- cian to r einterpr et sp e ctr al me asu r ement s otherwise than in “Wiener- Khinchin” terms. [...] op er ations me ant to me asu r e the Wiener-Khinchin sp e ctrum may unvoluntarily me asure something else, to b e r eferr e d to as the “c onditional sp e ct r u m” of a “c onditional ly c ovarianc e stationary” r andom function. [17] T aking the t wo pap ers [16,17] together w e can see that Mandelbro t expanded on this visio n b y discus sing sev er al fra ctional renew a l models, including in [16] a three state, explicitly nonstationa ry mo del with waiting times whos e probabilit y density function decay ed as a power law p ( t ) ∼ t − (1+ θ ) . This sto chastic pro cess was intended as a “ carto on” to mo del intermittency , in which “o ff” p erio ds o f no activity were interrupted by jumps to a negative (or positive) “on” a ctiv e state. His k ey finding, confirmed in [17] for a mo del with an a rbitrary n umber o f discrete levels, was that the traditional Wiener-Khinchine sp ectral diagno stics would return a 1 /f per iodo g ram and thus a sp ectral “infra r ed catastro phe” when view e d with traditional metho ds, but building o n the notion of conditional 6 stationarity prop osed in [15], that a co nditional p ower sp ectrum S ( f , T ) could be decompo sed in to a stationary part in which no ca tastrophe w as seen, and one depe nding o n the time ser ies’ length T , m ultiplying a slowly v ar ying function L ( f ). He found S ( f , T ) ∼ f θ − 1 L ( f ) Q ( T ) (3) where Q ( T ) T 1 − θ was slowly v arying, a nd tha t the conditional s p ectral density S ′ ( f , T ) ob eys S ′ ( f , T ) = d d f S ( f , T ) ∼ f θ − 2 T θ − 1 L ( f ) (4) Rather than repr esen ting a true sing ularit y in p ow er at the lowest freq ue nc ie s , in the Selecta [18] he desc r ibed the a pparen t infrared catastro phe in the p o wer sp ectral density in the fra c tional renewal mo dels as a “ mirage” resulting from the fact that the moments o f the mo del v aried in time in a step-like fa s hion, a prop erty he ca lled “conditional cov ariance stationarity”. In [17] Mandelbrot noted a clear contrast b et ween his co nditionally station- ary , non-Gaussian fractiona l renewal 1 /f mo del and his stationary Gauss ian fGn mo del (the 1 968 pap er conc e r ning which, with V an Ness, was then in press at SIAM Review): Se ction VI [... of this p ap er... ] showe d that s ome f θ − 2 L ( f ) noises have a very err atic sampling b ehavior. Some other f θ − 2 noises ar e Gaussian and, ther efor e, p erfe ctly “wel l-b ehave d;” an example is pr ovide d by the “fr actional white noise” [i.e. fGn] which is the formal deriva t ive of the pr o c ess of Mandelbr ot and V an Ness [i.e. fBm] He identified the o rigin of this erratic sampling be haviour in the non-er godic ity of the fractional renewal pro cesses. Niemann et al [25] ha ve recen tly given a very precise a na lysis of the b eha vio ur of the ra ndom pr efactor S ( T ) , obtaining its Mittag-Leffler dis tr ibution and chec king it b y simulations. 3 The Hurst effect vs. 1/f vs. LRD Informed in part by the a bov e historic a l in vestigations, the purp ose of this sec- tion is now to distinguish c o nceptually betw ee n 3 pheno mena which are still frequently e lide d. T o reca p, the phenomena a re: – The Hurst effect: the observation of “ anomalous” growth of ra nge in a time series using a diagno stic such as Hurst and Mandelbrot’s R/S or detrended fluctuation a na lysis (DF A)(e.g. [9,3]). – 1 /f noise: the observation of s ingular low frequency b ehaviour in the empir- ical p erio dogram of a time series . – Long range dep endence (LRD): a prop erty o f a s tationary mo del by c on- struction . This can only b e inferr e d to b e a prop ert y of an empirical time series if certain additional conditions a re known to be met, including the impo rtan t o ne of stationar it y 7 The reason why it is necessa r y to unpic k the relationship betw een these ideas is that ther e are three commonly held mispe rceptions ab out them. The first is that observation of the Hurst effe ct in a time serie s ne c essarily impl ies stationary LRD. This is “well known” to b e err oneous, see e .g . the work of [5] who show ed the Hurst e ffect a rising fro m an impos e d trend rather than from stationary LRD, but is nonetheless in pra ctice still not v er y widely appreciated. The se c ond is that observation of the Hurst effe ct in a time series ne c essarily implies a 1 / f p erio do gr am. Although less “well known”, [8], for ex ample, have shown an example wher e the Hurst effect a rose in the Lorenz mo del which has an exp onent ia l pow er sp ectrum rather than 1 /f . The thir d is the ide a that observation of a 1 /f p erio do gr am ne c essarily implies stationary LRD. As noted ab o ve, this is a mo re subtle issue, and a lthough little appreciated s inc e the pione e ring work of [15,16,17] it ha s now beco me central to the inv estiga tion o f w ea k er godicity breaking in physics. 3.1 The Hurst effect The Hurs t effect was o r iginally obser v ed as the growth of ra ng e in a time series, at first the Nile. The original diagnostic for this effect w a s R/S. Using the notation J (not H ) fo r the Joseph (i.e. Hurst) exp onent tha t Mandelbrot latterly advocated [19], the Hurst effect is seen when the res caled range[3,9] grows with time a s R S ∼ τ J (5) in the case tha t J 6 = 1 / 2. During the p erio d b e tw e e n F eller’s pro of tha t an iid s tationary pro cess had J = 1 / 2, and Ma ndelbr ot’s pap e r s of 1 965-68 on long range dependence in fGn [9], there was a co ntrov ers y a bout whether the Hurst effect was a co nsequence of no ns tationarity and/or a pre-a symptotic effect. This contro versy ha s never fully subsided [9] b ecause Occa m’s Razo r frequently fav ours at least the p ossibility of change p oints in an e mpir ically determined time series (e.g. [24]), and b ecause of the (a t first sig h t sur pr ising) no n-Marko vian prop erty of fGn. A key po in t to appreciate is that it is easier to generate the Hurst effect ov er a finite sca ling range, as mea sured for example by R/S , than a true wideband 1/f sp ectrum. [8] fo r example shows how a Hurst effect can appe ar o ver a finite range even when the p ow er sp ectrum is known a priori no t b e 1 /f , e.g . in the Lorenz a ttr a ctor case where the low frequency s p ectrum is exp onential. 3.2 “1/f” sp ectra The term 1 /f spec tr um is usually used to deno te p erio dograms whe r e the spe c - tral density S ′ ( f ) has a n in verse p o wer law for m, e.g. the definition used in [16,17] S ′ ( f ) ∼ f θ − 2 (6) 8 where θ r uns betw een 0 and 2 . One needs to disting uish here betw ee n b ounded and un b ounded pro cesses. Brownian, and fra ctional Brownian, mo tion are unbounded, nons tationary ran- dom walks and one can view their 1 / f 1+2 J sp ectral density as a direct conse- quence of nonstationar it y , a s Mandelbrot did (see pp 7 8-79 of [18]). In many ph y sical c o n texts how ever, such a s the on-off blinking qua n tum dot pro cess[2 5] or the river Nile minima studied by Hurs t[9] the signa l amplitude is a lw ays bo unded and does not g ro w in time, requiring a different explanation that is either stationar y or “conditionally stationary” . Mandelbrot’s b est known model for 1 /f no ise remains the stationa r y , er- go dic, fractional Gaussian noise (fGn) that he advoc a ted so ener g etically in the 1960s . But, evidently himself aw are that this had had received a disprop ortion- ate amount of attention, he was at pains late in his life (e.g. Selecta V olume N [18] p.20 7 , intro ducing the reprinted [16,17]) to stress that: Self-affinity and an 1/f sp e ctrum c an r eve al themselves in sever al quite distinct fashi ons ... forms of 1/f b ehaviour that ar e pr e dominantly due to the fact that a pr o c ess do es not vary in “clo ck time” but in an “intrinsic time” that is fr actal. Those 1/f noises ar e c al le d “sp or adic” or “absolutely intermittent”, and c an also b e said to b e “dustb orne” and “ acting in fr actal time”. He clearly distinguis he s the LRD statio nary Gaussian mo dels like fGn from from his “conditionally stationa ry” fractal time pr ocess , noting also that: Ther e is a sharp c ontr ast b etwe en a highly anomalous ( “ non-white”) noise that pr o c e e ds in or dinary clo ck time and a noise who se princip al anomaly is that it is r estricte d t o fr actal time. In practise the main imp o rtance of this is to caution that, used on its own, even a v ery so phis tica ted approach to the pe r iodog ram lik e the GPH metho d [3] cannot tell the difference betw een a time series b eing stationa ry LRD of the fGn t y p e and “just” a “1/f” noise, unless independent information ab o ut stationa rit y is als o av ailable. One route to reducing the am big uit y in future studies of 1 /f is to develop non-stationar y extensio ns to the Wiener-Khinchine theorem. An imp ortant step [12] has b een to distinguish betw een one which rela tes the sp ectrum and the ensemble av era ge co r relation function, and a seco nd relating the spectrum to the time a verage corr elation function. The imp ortance of this distinction can b e seen by c o nsidering the F ourier inv ersio n of the power s pectrum-do es the inversion yield the time or the ensemble av erag e? [E. Bar k ai, p ersonal co mm unication]. 3.3 LRD My rea ders will, I hope , now be able to s ee why I b eliev e that the commonly used sp ectral de finitio n of LRD has caused mis under standings. The problem has be en 9 that on its own a “1/f” b eha vio ur is necessa r y but not sufficient, and stationarity is also es s en tial for LRD in the sense so widely studied in statistics co mm unity (e.g. in [2] and [3]). One may in fact argue that the more crucial a spect of LRD is th us the “lo ose” o ne e mbo died in its na me, rather than the formal one embo died in the sp ectral definition, b ecause a 1 / f sp e ctrum c an only b e synonymous with LRD when ther e is an infinitely long p ast . The fact that fGn exhibits LRD by c onstruction becaus e the sta tionarity prop erty is assumed, and also shows 1 / f noise, and the Hurst effect has led to the widespread misconception that the conv erse is tr ue , and that o bserving “1/f” spectr a and/or the Hurs t e ffect m ust imply LRD. 4 Conclusions Unfortunately [17] received far less cont emp orar y attention than did Ma ndel- brot’s pap ers on heavy tails in finance in the early 1960 s or the serie s with v an Ness and W allis in 1 968-69 o n stationa ry fra ctional Gaussian mo dels fo r LRD, gaining o nly ab out 20 citations in its first 20 y ear s. This has been rectified sinc e but I b eliev e the consequences ha ve b een la sting. Perhaps it w as beca us e his work on the AFRP w as communicated primarily in the (IEEE) jour nals and conferences of telecommunications a nd computer science, that it w as lar gely in- visible to the con temp orar y audience that encountered fGn a nd fBm in SIAM Review and W ater Resourc e s Resea rc h. In fact, so invisible was it that one his most ar ticulate critics, hydrologis t, Vit Kleme ˇ s [11] used an AFRP mo del as a paradigm for the absenc e of the type o f LRD seen in the statio na ry fGn mo de l, clearly unaw a re o f Ma ndelbrot’s work. Kleme ˇ s’ pa per remains very worth while reading even to day . It s ho wed how at least t wo o ther classes o f mo del could exhibit the Hurst effect, the AFRP cla ss and also integrated pr ocess e s, such as AR(1) with a φ parameter close to 1. F ascinatingly , despite the fact that Man- delbrot had colleagues s uc h as the hydrologist W a llis who published in W ater Resources Resear c h, and thus may well hav e seen the pa per, if he did he chose not to enlighten Kleme ˇ s ab out his earlier work. Sa dly Kleme ˇ s a nd Mandelbro t seem also not to hav e subsequently debated nonstationary approa c hes on an equal fo oting with fGn, as with the adv antage of histor ical distance one can see the importa nce of b oth as non-ergo dic and er godic solutions to the 1 /f paradox. Although he revis ited the 1963 -67 fractional r enew al pap ers with new com- men ta ries in the v olume of his Selecta [18] that dea lt with m ultifractals and “1 /f ” noise , Mandelbro t himself neglected to mention them explicitly in his po pular histo rical account of the ge ne s is of L RD in [20]. Tha t he saw them as a r epresentin g a different strand of his work to fractional Brownian motion is clear from the way that fBm and fGn and the Ga ussian paths to 1 /f were each allo cated a separa te Selecta volume [19]. Despite the Selecta, the r elativ ely low visibility has r emained to the present day . Ma ndelbrot’s fractiona l renewal pa- per s are for example not cited or discuss e d ev en in encyclop edic b oo k s on LRD such a s Beran et al’s [3]. 10 The long term cons e q uence o f this in the ph ysic s and statistics literatures may hav e been to empha sise er godic solutions to the 1 /f problem at the exp ense of non- ergo dic ones. This seems to me to be imp ortant, b ecause, for example, Per Bak’s paradig m of Self-Org anised Criticality , in which stationary spec tra and correla tion functions pla y an essen tia l r o le, c o uld not sur ely have b een p ositioned as the unique so lution to the 1 / f problem [30] if it had b een widely r ecognised how differe n t Ma ndelbrot’s t wo existing routes to 1/ f already were. A cknow le dgements. I would like to thank Rebec c a Killick for in viting me to talk at ITISE 2016 , and helpful comment s on the manuscript from Eli Bark ai. I also gratefully acknowledge many v a luable discussions ab out the history of LRD and weak ergo dicity break ing with Nic k Moloney , Christian F r a nzk e, Ralf Metzler , Holger Kantz, Ig or Sokolov, Rainer Klage s , Tim Graves, Bobby Gr amacy , An- drey Cherstvy , Aljaz Go dec, Sandra Chapman, Tho rdis Thor arinsdottir, Kr istof- fer Rypdal,Mar tin Ryp dal, Bo gdan Hnat, Daniela F r oemberg , and Igor Goyc huk among man y other s. 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