APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind., 2001; 17:293}306 (DOI: 10.1002/asmb.440) The range inter-event process in a symmetric birth}death random walk Pierre Vallois and Charles S. Tapiero* Department of Mathematics, University of Nancy I, B.P. 239-54506 Vandoeuvre les Nancy Cedex, France ESSEC, B.P. 105-95021 Cergy-Pontoise, France SUMMARY This paper provides new results for the range inter-events process of a birth}death random walk. Motivations for determining and using the inter-range event distribution have two sources. First, the analytical results we obtain are simpler than the range process and make it easier, therefore, to use statistics based on the inter-range event process. Further, most of the results for the range process are based on long-run statistical properties which limits their practical usefulness while inter-range events are by their nature &short-term' statistics. Second, in many cases, data on amplitude change are easier to obtain and calculate than range and standard deviation processes. As a results, the predicted statistical properties of the inter-range event process can provide an analytical foundation for the development of statistical tests that may be used practically. Application to outlier detection, volatility and time-series analysis is discussed. Copyright 2001 John Wiley & Sons, Ltd. KEY WORDS: range process; inter-range events process; R/S analysis 1. INTRODUCTION Study of the range process is an important and often neglected topic of research compared with say, a process variance [1, 2], currently motivated by the application of ARCH, GARCH models and long run memory models [3}5]. Feller as early as 1951 remarked that it is di$cult to compute the range distribution in a symmetric random walk. Vallois [6}8], Vallois and Tapiero [9}13], Tapiero and Vallois [14] have renewed interest in this process and have derived some statistical properties of the range process for some speci"c discrete and continuous processes. For example, Vallois [6] has obtained results for discrete symmetric and asymmetric random walks as well as for continuous random walks. Particular characteristics such as the mean and the variance of these processes have been calculated by Vallois and Tapiero [9, 11]. Vallois [11, 12], has also calculated the probability distribution of the run length distribution of the range process which * Correspondence to: Charles S. Tapiero, ESSEC, BP 105-9502 Cergy-Pontoise, France Contract/grant sponsor: France-Israel ARIEL project Contract/grant sponsor: CERESSEC Published online 22 June 2001 Copyright 2001 John Wiley & Sons, Ltd. Received 1 January 2000 Revised 12 June 2000 294 P. VALLOIS AND C. S. TAPIERO has a complex form. Applications have also been pointed out with respect to range to scale (R/S) and Hurst exponent analysis. Hurst [15] who used a range to scale statistical analysis, also coined R/S analysis (see also Tapiero and Vallois [14]) used to characterize the &persistence' of time series (also called the Joseph e!ect by Mandelbrot and Wallis [16]). Since then, Mandelbrot and other researchers have expanded and published numerous papers on the R/S statistic (see also References [16}23]). There are as well many applications in "nance and economics (for example References [24}35]). Additional applications to outliers and volatility change detection are currently being conducted, using the analytical properties of the range and inter-range event processes derived here. This paper provides new results on the range inter-events process of a birth}death random walk. Motivations for determining and using the inter-range event distribution have two sources. First, the analytical results we obtain are simpler than the results for the range process and make it easier therefore to use statistics based on the inter-range event process. Further, most of the results on the range process are based on long-run statistical properties which limits their practical usefulness while inter-range events are by their nature &short-term' statistics. Second, in many cases, amplitude change data are easier to collect than range and standard deviations of processes. As a result, the predicted statistical properties of the inter-range event process provides an analytical foundation for the development of practical statistical tests. The results obtained here include the moments, the Laplace transform and probability distribution of the inter-range event process in a birth}death random walk. These results complement known results on the range and the extremely large literature of ARCH and GARCH modelling. Of course, results on simple random walks are derived as special cases. 2. THE RANGE INTER-EVENT PROCESS Let the time series x , t*0, de"ned by the following symmetric birth}death process: R x "x # , x "0, t*1 R> R R #1 w.p. p " 0 w.p. r R !1 w.p. q (1) (2) where is a Bernoulli random variable with parameters (p, q, r), with p'0, q'0, r*0, R p#q#r"1. If pOq, we have as a special case the asymmetric random walk while for p"q we have a symmetric birth}death process. The range process R , t*0 is de"ned by R R "Max [x , x , x , 2 , x ]!Min [x , x , x , 2 , x ] (3) R R R Since R , t*0 is non-decreasing in t, we can equivalently, study its inverse process (or the range R run length process). This is the "rst time that the range process reaches the amplitude a: (a)"Inf (t*0; R *a) (4) R Clearly, R (a"(a)'n) and therefore R can be studied instead through the run length (see L R References [7, 9}14]). We shall consider the range inter-event time distribution or the process (a)" (a#1)!(a). Assume that at time (a), the amplitude is a and that x '0. The interevent time F? Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 295 SYMMETRIC BIRTH}DEATH RANDOM WALK (a) is de"ned as the absorption time of the process at either x #1 or at x !a!1. In these F? F? two cases, the amplitude increases by a unit. Let ¹(a, b) be this absorption time: ¹(a, b)"Inf t'0; x "a or b R To determine the statistical properties of (a) we use the following lemmas. (5) ¸emma 1 Conditionally on x '0, F? B ¹(1,!a!1) (a)" (6) B ¹(!1, a#1) (a)" (7) Further, conditionally on x (0, F? In case of a symmetric birth}death random walk, i.e. p"q: B ¹(!1, a#1) " B ¹(1,!a!1) (a)" (8) Proof. The proof follows from the de"nition of the range inter-event process with an amplitude growth of a unit. Equation (8) follows directly from the symmetry property of the birth}death random walk. ) In the symmetric case we set p"q". Then r"1!2 and 3 ]0, 1/2]. ¸emma 2 Let a and k be "xed and x "x !x , n*0 be the process position change after time L L>F? F? (a). Then, (i) For x '0 F? (a#k)!(a)"Inf n*0; Max x!Min !a, Min x "a#k G G 0)i)n 0)i)n For x (0 F? (a#k)!(a)"Inf n*0; max a, Max x! Min x "a#k G 0)i)n G 0)i)n (9) (10) (ii) Let J "(x , u)(a)) be the algebra generated by the symmetric birth}death random F? S walk process with parameters (, , 1!2) up to time (a). (x !x ; n*0) is a random walk independent of J , thus, (a#k)!(a) is L>F? F? F? statistically independent of J and further, independent of x '0 (resp. x (0). F? F? F? Proof. We consider "rst the case x '0 and de"ne the extreme statistics: F? xN " Max x , x " Min x L 0)i)n G L 0)i)n G Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 296 P. VALLOIS AND C. S. TAPIERO Then "a#k (a#k)!(a)"Inf n*0; xN !x L>F? L>F? xN "Max xN , x ,x , ,x L>F? F? F?> F?> 2 F?>L "Max xN , x #x , x #x , 2 , x #x F? F? F? F? L "Max xN , x # Max x "x # Max (x ) F? F? 1)i)n G F? 1)i)n G since x "xN on x '0. Similarly, F? F? F? x "Min x , x # Min x "x #Min !a Min x F? G L>F? F? F? 1)i)n G 1)i)n which is the case since x !x "x !xN "!R "!a F? F? F? F? F? The case where x (0 follows by symmetry. F? Note that (x ; n*0) is a symmetric birth}death random walk with parameters (, , 1!2) and L independent of J "(x , u)(a)). Noting that (!x ; n*0) has the same distribution as (x ; F? S L L n*0), we deduce the independence of (a#k)!(a) and J . In particular, (a#k)!(a) is F? independent of the event x '0. ) F? Lemma 2 implies the following Lemma 3 as well. Lemma 3 In a symmetric birth}death random walk, two consecutive range interevents (a) and (a#1) are statistically independent. Proof. The proof is a consequence of the independence of (a) and J as well as (a#1) F? and J which implies the independence of (a) and (a#1). ) F?> With the above results, the moments of (a) are easily calculated. Using the results of Siebenaler and Vallois (1996), we prove the following proposition: Proposition 1 For a birth}death random walk with parameters (, ,1!2) 1 E( (a))" (a#1) 2 (11) (a#1) (a#2a#3(1!2)) a(a#1) (a#2) (a#1) (1!2) Var ((a))" " # 12 12 4 (12) and Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 SYMMETRIC BIRTH}DEATH RANDOM WALK 297 Proof. Recall that 1 a(a#1) E((a))" 2 2 and a(a#1) [(a!1)(a#2)#6(1!2)] Var( (a))" 48 First note that E((a))"E((a#1))!E((a)) 1 a(a#1) a#1 1 (a#1) (a#2) ! " " 2 2 2 2 2 The variance is calculated using the independence of (a) and (a): Var ((a)#(a))"Var ((a))#Var ((a))"Var((a#1)) ) Simple manipulations lead to (12). Note that the inter-event distribution is a function of the current state-amplitude. Further, because of the independence property (Lemmas 2 and 3), the correlation between two successive range growth events are necessarily null. As a result, the range growth inter-event process has its "rst two moments fully characterized by Proposition 1. This proposition implies that E( (a)) and Var ((a)) are increasing functions of the amplitude. In other words, over time, as a process evolves, the range growth rate is increasingly smaller. Asymptotically, the standardized variable has a mean which is decreasing in amplitude a, since, E( (a))/(Var ((a))+(3/(a. Of course, in the long run, when the inter-event time is large, its variance has little use. This analysis also raises some doubts regarding the validity of R/S analysis based on extremely large samples. In such cases, the range remains stable while the standard deviation grows constantly as the sample size increases. In this sense, some of the long-run statistics (such as in R/S analysis) have little use since beyond a certain amplitude as the time between range changes will be extremely large. We now generalize these results by considering the "rst two moments of I(a)" (a#k)!(a). Proposition 2 For a symmetric birth}death random walk model with parameters (, , 1!2), ((a#k) (a#k#1)!a(a#1)) k(2a#1#k) E(I(a))" " 4 4 Copyright 2001 John Wiley & Sons, Ltd. (13) Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 298 P. VALLOIS AND C. S. TAPIERO and 1 Var (I(a))" (a#k!1) (a#k)(a#k#1)(a#k#2) 48 1 (1!2) (a#k)(a#k#1) ! a(a#1) [(a!1)(a#2)#6(1!2)] (14) # 48 8 Proof. Equation (13) is proved as in Proposition 1, by using the expected value of the range inverse process. The inter-events variance is more di$cult to derive, however. To do so, we use the following relationship: L n(n#1)(n#2)(n#3) i(i#1) (i#2)" 4 G We also note that I\ (a#k)!(a)" [(a#j#1)!(a#j)] H which is the sum of statistically independent increments ((a#j#1)!(a#j)). Thus, I\ I\ Var ((a#k)!(a))" Var((a#j#1)!(a#j))" Var ((a#j)) H H 1!2 I\ 1 I\ [(a#j)(a#j#1)(a#j#2)]# (a#j#1) " 4 12 H H Further development leads to Equation (14). ) Again, using the independence property, we can calculate the Laplace transform of the range inter-event process summarized by Proposition 3. Proposition 3 Consider the birth}death random walk (, , 1!2) and let I(a)"(a#k)!(a), be the kth range growth event given an amplitude a. Then I(1#?) (1#?>) ¸* (s)"E(e!sI(a))" , s*0 I? (1#?>I)(1#?>I>) (15) 2!1#eQ#((s) (s)" , (s)"(1!2!eQ)!4 2 (16) where Proof. The proof is based on the Laplace transforms of (a) calculated by Vallois [7, 8] with "1/2 as well as Siebenaler and Vallois (1998) to which we add the independence property proved in Lemma 3. In Vallois [8], it was shown that 2(1#)? ¸* (s)"E(e!s(a))" , s*0 ? (1#?)(1#?>) Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 SYMMETRIC BIRTH}DEATH RANDOM WALK 299 with de"ned by the proposition. Note that (a#k)"I(a)#(a) Due to the independence of I(a) and (a) an application of the Laplace transforms operator leads to ¸* (s)"¸* (s)¸* (s) ?>I I? ? and therefore, by substitution, we obtain, Equation (15). ) The Laplace transform in this case is di$cult to handle; however, whereas the mean and variance of (a) can be computed from the "rst two derivatives of the Laplace transform, the explicit calculation is very unweildy. Further, although we shall be able to obtain the probability distribution of (a), we are not yet able to derive the probability distribution of I(a). Using the key observation (8) and a procedure suggested by Cox and Miller [36, p. 29] it is possible to calculate directly the probability distribution for the inter-event process (a)"(a#1)!(a). Proposition 4 Consider the birth}death random walk (, ,1!2) and let (a)"(a#1)!(a) be the time between two subsequent range growth events (a, a#1). Then 2 ?> [sin (i) L\] for 1)n)a P((a)"n)" G a#2 G 4 ?> P((a)"n)" [sin (i) L\] for n*a#1 G a#2 G G (17) (18) where " , "1!2 (1!cos (i)) a#2 G (19) Proof. The proof follows a procedure indicated by Cox and Miller [36, p. 29] for a random walk with two absorbing barriers that we adapt to our needs (although the formula which is given by Cox and Miller has, in fact, an error which we corrected). Let x "j and consider the symmetric birth}death random walk with parameters (, , 1!2). Then the inter-event time distribution is given by the following two-point absorption process f (n)" H\?\ P(¹(!a!1, 1)"nx "j): f (n)" H\?\ f (n!1)#f (n!1)#(1!2) f (n!1), !a!1(j(1, n*1 H>\?\ H\\?\ H\?\ 0 if n*1, j"!a!1 or j"1 1 if n"0 The solution of this equation (in a more general case) is pointed out as stated above by Cox and Miller [36, p. 29] but with a mistake. Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 300 P. VALLOIS AND C. S. TAPIERO Corollary 1 For a pure symmetric random walk, i.e. p"q"1/2: "cos (i) G and therefore, the inter-event probability distribution is (20) 1 ?> [sin (i)cosL\(i)] 1)n)a (21) P((a)"n)" a#2 G 2 ?> P((a)"n)" [sin (i)cosL\(i)] for n*a#1 (22) a#2 G G The cumulative distribution for the symmetric random walk is given in the following corollary. Corollary 2 For a pure symmetric random walk 1 ?> [(1#cos (i))(1!cosL(i))] 1)n)a P((a))n)" a#2 G P((a))n)"P((a))a) (23) 2 ?> # [(1#cos ((2j!1))) (cos?((2 j!1))!cosL((2j!1)))] a#2 H While, for a symmetric birth}death random walk, i.e. p"q (24) 2 ?> 1! L G , 1)n)a P((a))n)" [(sin (i)) (25) a#2 1! G G ? ! L 4 ?> H\ sin((2j!1)) H\ (26) P((a))n)"P((a))a)# 1! a#2 H\ H Recall that is given by (19). The proof is straightforward and uses summation of a truncated G geometric function. These distributions can be used for the construction of statistical tests for outlier and volatility detection. A discussion of the approach is included at the end of the paper. Remark 1 When n*a#1, we have for a pure symmetric random walk: 2 P((a)"n)" a#2 ?> ?> (cos (k))L\! (cos (k))L> II II (27) and 2 P((a)*n)" a#2 Copyright 2001 John Wiley & Sons, Ltd. ?> (cos (k))L\#(cos (k))L II (28) Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 301 SYMMETRIC BIRTH}DEATH RANDOM WALK Finally, additional analysis for the pure random walk leads to surprising simpli"cation of the inter-range process distribution. This is an essential result of this paper and it is summarized by the following proposition. Proposition 5 For 1)n)a and p"q"1/2, (i) If n is even, then P((a)"n)"0. If n is odd, then 1 P((a)"n)" CK , n"2m#1 (m#1)2K> K (30) (ii) If n is even P((a)'n)"P((a)'n!1). If n is odd, CK P((a)'n)" K , n"2m!1 2K (31) Proof. To prove Proposition 5 we shall require the following relationships that we state without proof. We note that "/(a#2) and set the following: Let m be a whole number, then (a) ?> !1 if m is not a multiple of a#2 cos (2km)" a#1 otherwise I ?> (b) cos (k)"0, I ?> (c) cos K\(k)"0, I ?> (d) let 0)m)a#1, then cosK (k)"!1#(a#2)CK /2K . K I With these equalities on hand, we turn to the main proof of the proposition. (i) Let 1)n)a. We have ?> ?> 1 (cos k)L\! (cos k)L> P((a)"n)" a#2 I I If n is even, then by (c), the sum of the two terms is null and therefore, P((a)"n)"0 Now assume that n is odd, and set n"2m#1. We have, m)2m)2m#1"n)a)a#1. We can apply (d) with 1 ?> ?> P((a)"n)" (cos k)K! (cos k)K> a#2 I I 1 (a#2)CK (a#2)CK> K! !1# K> P((a)"n)" !1# a#2 2K 2K> Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 302 P. VALLOIS AND C. S. TAPIERO and CK> CK P((a)"n)" K! K> 2K 2K> However, since CK CK> (2m)! (2m#2)! 1 (2m)! 1 K! K>" ! " 2K 2K> 2K(m!) 2K>((m#1)! ) 2 2K(m! ) m#1 (ii) We compute now the distribution function of (a). If n is even, P((a)'n!1)"P((a)*n)"P((a)'n)#P((a)"n)"P((a)'n) Now assume that n is odd, n"2m!1. Then P((a))n)"P((a))2m!1)" P((a)"i) 1)i)2m!1 If i is even, P((a)"i )"0, thus if i"2s#1, P((a))n)" P((a)"2s#1)" 0)s)m!1 0)s)m!1 CQ CQ> CK Q! Q> "1! K . 2Q> 2K 2Q ) To calculate P((a)"n) when n*a#1, it is necessary to evaluate the sums of type (c) or (d) but with k a whole odd number. We distinguish between the cases a even and a odd and summarize these results in the following Lemma 4 which is stated without proof. Lemma 4 (1) If a is even and s odd, then ?> cosQ (k)"0 II (2) If a"2b, let m be a whole number. Then ?> 0 if m is not a multiple of b#1 cos (2km)" (b#1) cos (2m) otherwise II (3) Let a"2b, s*1, s , s de"ned such that s"s #s (b#1), 0)s )b. Then ?> a#2 Q b#1 cosQ(k)" (!1)Q\H CQ>[email protected]>! CQ Q Q 2Q 2Q II H (4) Let a"2b. Then P((a)"n)"0 if n is even, n*a#1 2 K 1 P((a)*2m#1)" (!1)K\H CK>[email protected]>! CK K 2K 2K K H where m"m #m (b#1); 0)m )b. Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 303 SYMMETRIC BIRTH}DEATH RANDOM WALK These important results, unlike the previous distributions, indicates a no-memory property of the amplitude for pure random walks (which is an expected result however), i.e. the probability of a range growth is independent of the amplitude of the process. Further, the conditional probability of the amplitude increasing at the nth period, since the last amplitude growth, given that it has not increased up to that time, is given by (for n)a), P((a)"n) P((a)"n) " , h (n)" ? P((a)*n) P((a)"n)#P((a)'n) n"2m!1 (32) Elementary manipulations lead to the simple result, 1 (1/m2K\)CK\ K\ h (n)" " , n"2m!1 (33) ? (1/m2K\)CK\ #CK /2K 2m K\ K and therefore, the hazard rate, expressing the probability of an amplitude growth is inversely proportional to time: 1 h (n)" ; ? n#1 n odd, n)a (34) Thus, h (1)"1/2, h (3)"1/4, h (5)"1/6, etc. ? ? ? When n*a#1, the conditional probability of an amplitude growth is S (n!1)!S (n#1) ? h (n)" ? ? S (n)#S (n!1) ? ? (35) where > S (n)" (cos (k))L ? II An asymptotic analysis shows that the hazard rate function satis"es h (n) & 1!cos "2 sin(/2), "/(a#2) ? nPR (36) (37) which is constant. Of course, when the amplitude is large, the probability tends to zero as well. Other properties of the process can be calculated as well. Further, applications using statistical data on say intra-day trading data on the stock market can be used to compare such an analysis with ARCH-GARCH modelling and R/S analysis. 3. DISCUSSION AND APPLICATIONS Range and R/S analysis have attracted much attention in the economics and "nance literature, seeking to develop a statistical foundation for a non-linear statistical approach to "nance. Namely, constructing statistical tests for the detection of chaos and tests for the departure from the standard normal assumptions made in econometrics and "nance. Most studies have emphasized the need for large quantities of data since available statistical results were based on essentially long-run results. This is in particular the case for R/S analyses. Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 304 P. VALLOIS AND C. S. TAPIERO These studies are important to validate "nancial mathematical theory. This theory presumes both the &predictability' of future prices, interest rates as well as other and relevant time series to "nancial markets and processes or equivalently, uncertainty is expressed in terms of a Martingale. Such processes have been presumed by Bachelier already in 1900 and underlie the Random Walk Hypothesis in "nance. These latter processes have special characteristics and consist in continuous time and states of independent increments, independently and identically distributed Gaussian random variables with mean 0 and variance t. This means that the variance grows linearly over time. This facet of &the growth of uncertainty' has been severely criticized and numerous statistical tests have been based on it to demonstrate that the underlying process need not be the Brownian motion. In particular, it can be shown that stationary and independence of increments imply the functional relationship: f (t#s)"f (t)#f (s) and vice versa (where f ()) is a continuous function), while solution of this equation is uniquely given by the linear time function f (t)"tf (1). Empirical evidence has shown, however, that "nancial series are not always &well behaved'. They may exhibit unpredictable and &chaotic behaviour' which underscores the &non-linear science' approaches to "nance. Rather, in many cases, it is observed that data can behave sometimes &unpredictably' while at other times, it may exhibit regular variations. &Bursts' of activity, &feedback volatility' and broadly varying behaviours by stock market agents, &memory', etc., all contribute to processes which do not exhibit the Martingale property. Further, even aggregation of time series that are mildly auto-regressive can turn out to have long-run memory. The study of these time series has motivated a number of approaches falling under a number of themes spanning: heavy tails (or Pareto}Levy stable); Long-term memory and dependence; Chaotic analysis; Lyapunov stability analysis; complexity analysis; fractional Brownian motion; multifractal time series analysis and, more recently, R/S analysis indicated here. For example, the presence of long-run memory in say assets returns has important implications for many of the paradigms used in modern "nancial economics. For example, optimal consumption-savings and portfolio decisions may become extremely sensitive to the investment horizon if stock returns were long-range dependent. Problems also arise in the pricing of options and futures since the class of models used are incompatible with long memory. Traditional tests of the capital asset pricing model and the arbitrage pricing theory (APT) are no longer valid since the usual forms of statistical inference do not apply to time series exhibiting such persistence. Also, the conclusions of more recent tests of e$cient markets hypotheses or stock market rationality also hang precariously on the presence or absence of long-term memory [34]. Further, if speculative prices exhibit dependency, this structure would be inconsistent with rational expectations and would thus make a strong case for technical forecasting on stock prices (contrary to the conventional assumption that prices #uctuate randomly and are thus unpredictable). In other words, the Martingale approach to "nance and its many applications and results will be of little use. The underlying properties of R/S statistics which are related to memory and persistence (or antipersistence) can be traced to the relative robustness of a series standard deviation statistic compared to that of a range process based on the same series. For example, if the range increases over time much faster than the standard deviation, this may mean that the series has become chaotic (in the sense that the range that is sensitive to more recent data, changes much faster than the standard deviation which is more robust). These relative e!ects, if well understood, can be used pro"tably for detection and departure of the Gaussian-Random Walk hypothesis. This paper complements such analysis by deriving explicit results regarding the probability distribution of the inter-range event distribution. The distributions derived in this paper simpli"es Copyright 2001 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2001; 17:293}306 SYMMETRIC BIRTH}DEATH RANDOM WALK 305 the small-sample analysis of range-based statistical analyses, providing thereby an alternative to statistical studies based on range and R/S analyses. Further, inter-range growth data can be used for a number of problems of current importance in time series. Outlier detection, volatility analysis and the detection of chaos are such examples that will be considered in greater detail in subsequent research. The detection of outliers and a change of volatility are important and useful problems. Barnett and Lewis [37] (with the "rst edition in 1978) for example, point to over 600 tests that have been developed and applied to detection of outliers. In this sense, it is di$cult to presume that one and only one test can be used to detect an outlier. As a result, the inter-range event may provide an additional approach specially suited for time series. Similarly, ARCH and GARCH models are used extensively to estimate variances, processes of volatility and variations in volatility. The interest in these problems is both theoretical and practical. For example, in the valuation of options, knowledge of the variance of an underlying stock (or detection of a shift in this variance) can be extremely useful for better predicting the price of options. For outliers in iid samples, such an approach was pointed out by Irwin in 1925. Explicitly, for a standard deviation given by , Irwin proposed the following statistic (in Reference [37]) for testing upper outliers in a sample of size n were x is the ith ordered statistic: G W x !x X/ and W x !x X/ L L\ L\ L\ The inter-range event provides an equivalent statistic for time series but based on the time consumed between two (or more) consecutive changes in the range. 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