Proc. NatL Acad. Sci. USAVol. 79, pp. 4501-4505, July 1982Applied Mathematical Sciences

Fractal dimensionality of Levy processes(mean first-passage times/maxima moments)

V. SESHADRI AND BRUCE J. WESTCenter for Studies of Nonlinear Dynamics, La Jolla Institute, P. 0. Box 1434, La Jolla, California 92038

Communicated by Elliott Montroll, April 7, 1982

ABSTRACTT We determine the fractal dimensionalityD of thetrajectories of a class of translationally invariant Markov pro-cesses. We also provide two simple operational measures to esti-mate D.

space. The mean first-passage time, however, is not a sharplydefined quantity because the first-passage time distribution isvery broad (13, 14). Maxima moments, on the other hand, area sensitive measure of the fractal dimensionality.

Recently, considerable attention has been devoted to diversephysical phenomena exhibiting a clustered behavior in spaceor time domains (14). Examples of such clustered behaviorappear in processes in fluid mechanics (1-4), solid state physics(ref. 1, p. 16; ref. 5), astrophysics (ref. 9, p. 170), data trans-mission systems (ref. 1, p. 93), and Brownian motion (ref.. 1, p.201). There is apparently a close connection between such clus-tered behavior in space or time and the Hausdorff-Besicovitch(fractal; ref. 1, p. 16) dimensionality of these processes.The fractal or Hausdorff-Besicovitch dimensionality D of a

set may be defined as follows. If a finite part El of the set isdivided into N identical parts, each of which is geometricallysimilar to El with the similarity ratio r, then D = tn N/en(1/r).Analytic estimates of the fractal dimensionality of dynamicalprocesses has so far been carried out only for Brownian motion(ref. 1, p. 201). In this paper, we examine the fractal dimen-sionality of the trajectory of a class of translationally invariantMarkov processes and also provide two operational measuresfor estimating time.A simple measure of the fractal dimensionality of a model

dynamical process that exhibits clustering has been developedby Hughes et aL (10). The model is a discrete random walk ona lattice with transition probabilities drawn from a distributionwhich does not have a finite variance-i. e., a Levy distribution(11). Hughes et al. were able to associate the short-scale be-havior of the structure function to the fractal dimensionality ofthe walk. In the continuous space-time limit their model pro-cess reduces to a special case of the translationally invariantprocesses considered herein.The distribution functions of translationally invariant Markov

processes that we consider satisfy the Bachelier-Smoluchowski-Chapman-Kolmogorov (BSCK) chain conditionand are called LUvy processes (12). Levy processes in one, two,and three dimensions* are represented by their characteristicsfunction 4(Ilkl) - exp{-lklj}. The exponent. p. of the character-istic function determines the essential physical properties oftheprocess. In particular, we show that Ad is the fractal dimen-sionality of the trajectories of the process.

In this paper we propose the use of mean first-passage timesand maxima moments as operational measures of the fractaldimensionality of the trajectories of Levy processes. The ideaof using mean first-passage times as a measure of the fractaldimensionality has its seed in the heuristic connection, estab-lished by Mandelbrot, between the fractal dimensionality ofBrownian motion and the total time spent within a region of

Levy distributions

Translationally invariant stationary Markov processes in con-tinuous space are described by probability densities that satisfythe Bachelier-Smoluchowski-Chapman-Kolmogorov chaincondition,

P(X2 - x1,t) = J P(X2 - x, t - r) (x - x1,)dx [2.1]

where P(x2 - xj,t) is the probability that the value ofthe processX changes from xi to x2 in time t. The characteristic function4(k,t) is defined as the Fourier transform of the -probabilitydensity-i.e.,

Jx

0(k,t)= dx eia P(x,f) ._x~~~~~~~~~~~~~~~

[2.2]

For processes satisfying the chain rule (Eq. 2.1), #(k,t) obeysthe product rule.

4(k,t) = 4(kt - r) 4(kr). [2.3]

Montroll and West (12) noticed that, since dI(k,t) satisfies rule2.3, it is an infinitely divisible stable distribution (11).

The most general form of ck(kt) for infinitely divisible stabledistributions was obtained by L6vy (15) and Khinchine and Levy(16). For symmetric processes, the most general form (11) isgiven by

O(kt) = exp [-bt lkijI + icw(lc ).)k [2.4]

In Eq. 2.4, A., b, and c are constants obeying the restrictions0. pA-2, b .0 and -1 c . 1. The function w(k,pI) is de-fined by

w(k,p) = tan (irp/2) if p. # I

=-2 nlkbtlir

ifpA= 1. [2.5]

The class of processes whose characteristic functions satisfyEq. 2.4 are called Levy processes. The most important featureof Levy distributions is that, except for p. = 2, the distributionsP(x,t) do not possess finite moments of all orders. This can beseen most easily from the fact (12) that, for t > 0,

* We shall henceforth denote the Euclidean dimension either by thesymbol E or simply by the term "dimension." The fractal or Haus-dorff-Besicovitch dimension is always indicated either as such or bythe symbol D.

The publication costs ofthis article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertise-ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

4501

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

8, 2

020

4502 Applied Mathematical Sciences: Seshadri and West

lim P(xt) - 4btfr(A) sin (vri/2)/lrxM+l. [2.6]

Thus, all a moments defined byoe

(xIli) = J lxi" P(x,t)dx [2.7]

are finite for a < A and are infinite for a > ti. In particular,the variance is infinite.

In two and three dimensions we restrict our attention to cen-trosymmetric Levy distributions defined by the equation

4(k,t) = 4(1kl,t) = exp[-btlkitj

possible to construct simple evolution equations of the diffusiontype for the probability density-i.e., equations involvingPxt) . Such first-derivative equations, in general, contain in-at

tegral operators in the position variables. For example, in onedimension when pt # 2, the evolution equation can be shown(17) to be of the formaP(xt) bt

=- sin(nm/2) E(/ + 1)a3t r

i

[2.8]where IkI = (kx2 + ky2)1/2 in two dimensions and Ikj = (k 2 +ky + kz2ft'2 in three dimensions, respectively. The radially andspherically symmetric probability distribution functions in twoand three dimensions are defined, through their respectiveFourier transforms, via the relations

P(r,t) = 2-I e btkMo(kr)kdk; 2E,

and

P(r,t) = 2y7r e btk~sin krkdk; 3E, [2.10]

respectively. These L6vy distributions P(rt) are similar to theone-dimensional distribution P(x,t) in that for pA c 2 they alsodo not possess finite moments of all orders. It can be shown viaan asymptotic analysis similar to the one earned out by Montrolland West (12) that, for pA c 2, in two and three dimensions,P(r,t) has the asymptotic form

P(r,t) - (2)r bt r-E-t; E = 2,3. [2]

From Eq. 2.11 it is clear that, for A < 2, all moments of ordera defined by

(r") = (2r)E- f r"P(r,) rE1 dr; E = 2,3, [2.12]

are finite for a < pa and infinite for a 2The important feature of the asymptotic behavior (Eqs. 2.6

and 2. 11) of LUvy distributions for 0 s pA c 2 is the power-lawtail. From these asymptotic forms one can see that the proba-bility P(r > rj) that the process r > iq, for large values of q, is

(0

4) = (2 )E- j P(r t) rEl dr; E = 1,2,3,

1-~const- .

('c P(Y,t)x

cody{1 + c sign(y - x)}

x-Yp

In the next section, we demonstrate how formally simpler evo-lution equations possessing higher time derivatives can be ob-tained for certain values of the parameters A and c.

Differential evolution equations

Under certain conditions, differential evolution equations canbe obtained for the probability density which, in general, in-volve higher lime derivatives. 'We first consider the one-di-mensional case.

Let ;. be a rational number equal to m/n where m and n areintegers and the parameter c in Eq. 2.4 is selected to be zero.When m is even, we differentiate Eq. 2.4 successively n timeswith respect to time and inverse Fourier transform the resultingequation to obtain

[2.14]

anen+YWZ nEamP~x~f).m-P(xlt = (l)y+ /2bnmP(xt)a-1t" ax",

[3.1]

To ensure that the solutions of Eq. 3.1 are real and positive-i.e., they are probability densities-they must satisfy n initialconditions P(x,0), P(') (x~t)110...,Pn-'\x~t)1t=0 (the super-scripts denote the order of differentiation with respect to time).Though these sufficient initial conditions may be formally ob-tained from Eq. 2.4, they are, in general, as hard to evaluateas the inverse Fourier transform of the function O(k,t) itself.When m is odd, we differentiate Eq. 2.4 successively 2n

times and, after inverse Fourier transforming, obtaina271 a2m-P( ,t) = (-l)mb2" P(x,t) . [3.2]

Analogous to the even m case, solutions of Eq. 3.2 must satisfy2n initial conditions.When c # 0, it can be shown that closed differential equa-

tions can be obtained for pa = rn/n $ 1, with value of c givenby

[2.13] c = tan[irp/2nj]/tan[mv/2n];

Thus, the asymptotic behavior of the process r is that of a hy-perbolic random variable. As pointed out by Mandelbrot suchhyperbolic distributions preserve self-similarity and have tra-jectories with fractal dimensionality pA (ref. 1, p. 133). Thus,L[vy processes with exponent have trajectories with fractaldimensionality pA.

Even though the Levy distributions are characterized by ap-parently simple Eq. 2.4 and 2.10, there are twro major diffi-culties in understanding the physical properties of these dis-tributions (12). First, the probability density P(xyt) or P(rt) canbe evaluated in a closed form only for special choices of theparameters and c. Apart from the diffusion and the Cauchycases, a few other one-dimensional cases have been discussedby Zolotarev (see ref. 12). The second difficulty is that it is not

j- ±1, ±2, . a.0 p= +l, ±,...± r. [3.3]

The above construction can be extended to higher spatialdimensions.

Eqs. 3.1 and 3.2 are not particularly useful for computingP-x,t) itself because they have to be solved subject to compli-cated initial conditions. However, we have been able to usethem to compute properties such as the mean first-passagetimes and maxima moments. This utilization is discussed in thenext two sections.

First-passage times for lvy processes

Mandelbrot has discussed qualitatively the connection that ex-ists between the fractal dimensionality of the Brownian process

Proc. NatL Acad. Sci. USA 79 (1982)

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

8, 2

020

Applied Mathematical Sciences: Seshadri and West

and the average amount of time spent by the process in a givenregion of space. For persistent processes (ref. 9, p. 570; ref. 10)the average amount of time spent by the process in a given re-gion of space is infinite and a quantitative connection with thefractal dimensionality is hard to establish. A close quantitativeconnection exists, however, between the mean first-passagetime (for definitions, see refs. 9 and 14) to escape a given regionof space and the Hausdorff-Besicovitch dimensionality. Forexample, for Brownian motion (aL = 2) starting from the origin,the mean first-passage time T1 (4) to reach

T J1(4)= F(f4t) dt

- 42/n L 22±2/n cc1+2/n e l +2/1) I-'reo(2t + b

[4.4]

The most important feature of the behavior of the first-pas-sage time T1() is embodied in the first factor f2/n, 2/n -pu.The second factor is a monotonic increasing factor of n. Whenn = 1, we recover the result for the diffusion process (13)-i.e.,

i the points ± fii a circle of radius f

lE,2E,

iii a sphere of radius 4 3Eis proportional to g2 (13). Thus, the mean first-passage timescales with 4 with exponent 2 which is also the fractal dimen-sionality of the Brownian trajectories (in Mandelbrot's termi-nology, the Browmian line to E trail). We show below that meanfirst-passage times ofLvy processes also exhibit a similar char-acteristic scaling behavior with respect to 4. In fact, the expo-nent of 4 is precisely p. which suggests that the exponent of 4is the Hausdorff-Besicovitch dimensionality. We establish thisscaling relation by using exact boundary value techniques forp = 2/n, n = 1. oo and an approximate technique for ar-bitrary values of t.

Exact Mean First-Passage Times. In general it is difficultto calculate the exact mean first-passage times ofarbitrary Mar-kov processes. However, when the equation ofevolution for theprobability density is a differential equation possessing only aLaplacian in spatial variables, the first-passage problem can becast as a boundary value problem. Let us first consider one-dimensional processes.The exact first-passage time to reach the point x = 4 start-

ing from the origin, can be computed for processes which satisfy(a = 2/n):

n P(x,t) = ()n+l n xt) [4.1]

Letf(x,t) be the probability that the process has not crossed theboundaries x = + 4, given that it starts at the origin at time t= 0. From the theory of first-passage times (9) f(x,t) satisfiesEq. 4.1 with the boundary conditions fx,0) = 6(x) andf(±4,t)= 0. In addition, the n initial conditions on the derivatives off(x,t) must be specified to ensure thatf(x,t) is real and positive.The solution f(xt), using the procedure detailed in ref. 13, isthen given by

f(x,) =

1 (>2e +lxrx] [e {2e 1)124]exp L~~~~24 ~b [4.2]

In Eq. 4.2 only the real positive eigenvalue in the boundaryvalue problem has been chosen so thatf(x,) is real and positive.The cumulative distribution function F(4, t) is then

F(4, t) = dxf(x,t)

( [) (2 + 1ir2

2 -L 1)'xp _ bt .

bfreo' (2 f 1)- 2f

[4.3]

This result immediately yields the mean firstpassage time toreach 4,

T1(4) = 42/2b. [4.5]

The exponent 2 of 4 is the same as the fractal dimensionalityof the Brownian trail. When n = 2, we get the result for theCauchy process,

8GTI(f)= f b2 [4.6]

where G is the Catalan's constant (=0.915956 ...). The expo-nent of 4 is consistent with Mandelbrot's identification of thefractal dimensionality of the Cauchy trail as unity.The scaling behavior, proved above for one dimension, can

be extended to higher dimensions. For example, in two di-mensions the mean first-passage time T1(4) to a circle of radius4, and in three dimensions to a sphere ofradius 4, can be com-puted from the radially and spherically symmetric cumulativedistribution functions F(4,t) given in ref. 13, respectively. Thescaling result is Tj(t.) 4-b, 2= 2/n, as obtained above.

Approximate T1(4, An exact evaluation of Tl(4) for arbitraryp. is, in general, complicated for two reasons. First, if p. is ir-rational there are no differential evolution equations. Second,even if p. is rational but cannot be expressed as 2/n, the evo-lution equation possesses spatial derivatives of order higherthan 2. For such cases, the boundary value technique as usedabove no longer applies. Hencel we proceed by using an ap-proximation technique as follows.The approximation procedure rests on the assumption that

one can replace the conditional probability density f(xt) byP(x,t) in Eq. 4.3. Such a replacement yields the correct scalingbehavior ofT1(4) but not necessarily the correct coefficient. Fortransient processes, such a replacement yields a coefficient ofSF which is larger than the correct one but is finite. For per-sistent processes, the approximate coefficient turns out to beinfinite. To see this, consider the one-dimensional case

T1(4) = F(4,t) dt - A dt AfP(xt)dx .O o -{~~~

[4.7]

For a Levy process, we substitute for P(x,t) the Fourier trans-form of Eq. 2.4 to obtain in terms of the scaled variables x =x/4, k= k4, t = (bt)l/A,

T1(4) b- [4.8]

x [iOfccdrjcuf dke~fi'e-^(1 + @icok,p) )]

Thus, Eq. 4.8 has the correct scaling behavior. The coefficientof 4M, however, is finite only when p. K 1, as can be seen byexamining the singularities in the integrand of Eq. 4.8 after in-tegrating over t. The integral is finite only for transient pro-cesses (p. < 1) and is infinite for persistent processes (pL > 1)(ref. 9, p. 570; ref. 10).An entirely analogous argument can be presented for the two-

Proc. Natl. Acad. Sci. USA 79 (1982) 4503

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

8, 2

020

4504 Applied Mathematical Sciences: Seshadri and West

and three-dimensional cases. In two and three dimensions, wereplace F(tt) by

F( J,t) P(r,t) rE1 dr; E = 2,3. [4.9]

In both these cases the approximation procedure carried outabove yields the correct scaling behavior Tj(s) -M. The ap-

proximate coefficient of is finite for 0 g < 2 in two di-mensions but is finite for all ,a, 0 s g 2 in three dimensions.Thus, the approximation procedure can always be applied totransient processes.

The results obtained above are for continuous Markov pro-

cesses. Weiss and Rubin (18) have considered discrete randomwalk processes with transition probabilities with infinite vari-

ance and have obtained scaling results for mean first-passagetimes, similar to Eqs. 4.4 and 4.8. In the continuum limit theirresults agree with those given in this section. The followingpoint should be noted, however. In discrete random walkmodels, with transition probabilities with infinite variance,there exists a difference equation of evolution on the lattice. Inthe continuum limit of these models, no closed differentialequation with dUlt exists. This is precisely the reason for con-structing closed differential equations with higher order timederivatives (cf. Eq. 4.1).

Maxima moments for LUvy processes

A disadvantage of using the mean first-passage time as an in-dicator of the fractal dimensionality of a process is that it is nota sharp measure (14). An alternative property ofa Levy process

which is a sensitive measure of the fractal dimensionality is thedistribution of the maxima attained in a given time t (for a def-inition of maxima, see ref. 14). For Brownian motion (A.t = 2)all maxima moments are finite and well defined. However, thisis not the case for L6vy processes with 1t < 2. In this latter case,

maxima of certain order only are finite. This is to be expectedsince the distribution P(x~t) or P(rt) for ta < 2 has a power-lawhail.We define the fractional moment of order a of the maxima

distribution as

00

Za(t) = af df {a- [1 - f(4t)] . [5.1]

For Levy processes, Za(t) is finite iff a < ,a, IL < 2, and is infiniteotherwise. Equally interesting is the scaling behavior ofZa(t)i. e.,

Za(ttO ,t a<< U [5.2]

These results can be demonstrated in an exact fashion when theequation of evolution has the Laplacian form (Eq. 4.1).We first consider the one-dimensional case. Substituting the

expression from Eq. 4.3 into Eq. 5.1 and Laplace transformingboth sides, we obtain (p, = 2/n)

)'1 (2 t+ [5.3]

In terms of the scaled variable (6171)lM {we can-write Eq.S.3as

AZ (e) [5.4].-)=61+at J

4a t1' b J 0- a-I ( 1)7 (2 + IJ) I(24)A + (#2e 1)Y'L [5.5]

It is immediately clear from the inverse Laplace transform ofEq. 5.4 that

Za (t>= F( 1A [5.6]

The coefficient Aa, however, is finite only when a < A, as can

be demonstrated by performing the sum in Eq. 5.5 by a contourintegration.

In two and three dimensions the scaling ofZ(t) with t-i. e.,

Z,,(t) - t"-can be established by a similar procedure. Belowwe present a heuristic argument valid in each of the three di-mensions and also valid for all values of p., 0 -S < 2 to showthat A. diverges for a A

We proceed by splitting the integral Eq. 5.1 as

Zd,,(t)= af d [a-i[1 -F )zat

f+da d a- [1 -F (4t)] [5.7]

where F(4,t)-is the appropriately defined cumulative distribu-tion function in one, two; and three dimensions (cf. Eqs. 4.3and 4.9). In Eq. 5.7 -q is arbitrarily large. The first integral isfinite for any finite value ofBand the divergence of Z4x(t), ifany,arises from the second integral. For any fixed time t, for suf-ficiently large values of ?? we may replace [1 - F(4t)] by

JrE P(r,t) dr; E = 1,2,3. [5.8]

The asymptotic behavior of P(rt) given by Eqs. 2.6 and 2.11immediately leads to the result

lim [1 - F(4,t)1 - l/{ [5.9]

in each ofthe three cases. Substitution ofEq. 5.9 into the secondintegral of Eq.. 5.7 yields the result that maxima moments oforder a are finite iff a < ja, and are infinite for a : pk.

Conclusions

Herein we summarize our main results and discuss some oftheirphysical implications:

(i) Levy processes in one, two, and three dimensions obeythe asymptotic law P(r> 7v) T.Thus, the exponent is

the fractal dimensionality of the Levy trajectories.(ii) Differential evolution equations for the probability den-

sity have been obtained for L6vy processes with certain param-eter values. These evolution equations are, in general, not ofthe diffusion type and involve higher-order derivatives in spaceas well as time. For LUvy processes with second degree spatialderivatives, we have obtained exact mean first-passage timesand maxima moments.

(iii) The mean first-passage time to a boundary has beenshown to scale as in one, two, and three dimensions, 0 c

2. Thus, the scaling of the mean first-passage times can

serve as an operational measure of the fractal dimensionality ofthe process.

(iv) The maxima moment Z.(t)'in a time interval t has been

where

Proc. Nad Acad. Sci. USA 79 (1982)

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

8, 2

020

Applied Mathematical Sciences: Seshadri and West

shown to scale as tr/M in one, two, and three dimensions andare finite as long as a < ,' and are infinite when a 2 a, i <2. These moments are sensitive measures ofA and hence of thefractal dimensionality of the process.

(v) The fractal dimensionality of other properties of Levyprocesses can also be discussed in terms of the characteristicfunction. We present heuristic arguments for determining thefractal dimensionality of the zero-crossing set and the graphsof such processes.

Zero-Crossing Set. A simple scaling calculation shows thatthe probability of occupation of the origin of a Lvy process isP(0,t) - tC"Q The cumulative lime spent at the origin betweentimes 0 and t is proportional to t1-"t*, up to an additive constant.We associate the fractal dimensionality of the zero-set with thescaling of the time spent at the origin for t > 0-i.e., D = 1- 1/,. This association is valid for A 2. 1. This result is con-sistent with that of the Brownian process for which jt = 2, D= 1/2 and also that of the Cauchy process for which A = 1, D= 0. For all values of C< 1, D = 0.Graphs of Levy Processes. We argue that the fractal di-

mensionality of the graph X(t) vs. t for a one-dimensional Lvyprocess is D = 2 -1/1.4, au> 1. The argument is similar inspirit to the one presented by Mandelbrot, wherein he uses theLipschitz condition. Instead of the Lipschitz condition, weuse the result for the mean magnitude ofdisplacement for L6vygraphs, (fX(t) - X(0)1) - t"M, t > 0, A > 1. We note that thenumber of square boxes of side r required to cover the lengthof the graph between t and t + r is equal to r1/U'1area of the graph r \aor

--J2 The ttat number of boxes rearea of the box r /

quired to cover the graph between t = 0 and t = 1 is N =ruAM -

= r"'2. The fractal dimensionality is then D = log NIr

log (1/r) = 2 -11, A > 1/2. Thus, for the Brownian motion

/i = 2, D = 3/2 which agrees with known results. When Adecreases from 2 to 1/2, D decreases monotonically from 3/2to 0. For A -- 1/2, D = 0.

The authors acknowledge the many stimulating conversations withProf. Katia Lindenberg and Prof. E. XW. Montroll. The financial supportofthis research by the Office ofNaval Research and the Air Force Officeof Scientific Research is also acknowledged.

1. Mandelbrot, B. B. (1977) Fractals, For-r, Chance and Dimension(Freeman, San Francisco)

2. Frise, U., Sulem, P. L. & Nelkin, M. (1978)1J. Fluid Mech. 87,719.

3. Siggia, E. D. & Aref. H. (1980) Ann. N.Y. Acad. Sci. 367, 368.4. Ruelle, D. & Takens, F. (1971) Comm. Math. Phys. 20, 167.5. Monin, A. S. (1978) Sov. Phys. Usp. 21, 429.6. Rabinovich, M. 1. (1978) Sov. Phys. Usp. 21, 443.7. Scher, H. & Montroll, E. (1975) Phys. Rev. B 12, 2455.8. Scher, H. & Montroll, E. (1973)J. Stat. Phys. 9, 101.9. Feller, W. (1966) An Introduction to Probability Theory and Its

Applications (Wiley, New York), Vol. 2, 2nd Ed.10. Hughes, B. D., Schlesinger, M. F. & Montroll, E. W. (1981)

Proc. Natl Acad. Sci. USA 78, 3287-3291.11. Gnedenko, B. V. & Kolmogorov, A. M. (1954) Limit Distribu-

tionsfor Sums ofIndependent Random Variables (Addison-Wes-ley, Cambridge).

12. Montroll, E. WV. & West, B. J. (1979) in Fluctuation Phenomena,eds. Montroll, E. W. & Lebowitz, J. L. (North-Holland,Amsterdam).

13. Seshadri, V. & Lindenberg, K. (1980) J. Stat. Phys. 22, 69.14. Lindenberg, K., Shuler, K., Freeman, J. & Lie, T. J. (1975)1.

Stat Phys. 12, 217.15. LAvy, P. (1937) Theorie & 1 addition des variables altatoires

(Gauthier-Villars, Paris).16. Khinchine, A. Ya & Levy, P. (1936) Comptes Rendus 202, 274.17. Gelfand, I. M. & Vilenkin, N. Y. (1964) Generalized Functions,

Applications of Harmonic Analysis, translated by Feinstein, A.(Academic, New York), Vol. 4.

18. Weiss, G. H. & Rubin, R. J. (1980)J. Stat. Phys. 22, 97.

Proc. Nad Acad. Sci. USA 79 (1982) 4505

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

8, 2

020