Appendix 1A: Mathematical Properties of Lévy Stable Distributions. Consider the random variable < X <. The probability of obtaining X in the small

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1 1 I. CONTEXT Characteristic Functions. Gaussian Central Limit Theorem. Lévy Stable Central Limit Theorem. Laws of Large Numbers. Generating Lévy flights on computer. Appendix 1A: Mathematical Properties of Lévy Stable Distributions. II. CHARACTERISTIC FUNCTIONS Consider the random variable < X <. The probability of obtaining X in the small interval (x, xdx) is denoted with P X (x)dx. P X (x) is called a probability density function (PDF). In this text we assume that it is a smooth function. The PDF must satisfy the normalization condition P X (x)dx = 1, (1) and the non-negativity requirement P X (x) 0. Obviously if these conditioned are not satisfied P X (x) cannot be used used as a probabilistic measure. The n th moment of the random variable X is x n = and the variance is defined as x n P X (x)dx, (2) σ 2 = x 2 x 2. (3)

2 2 Not all PDFs have a finite variance, a counter example is the Cauchy or Lorentz PDF P X (x) = 1 π The characteristic function of the random variable X is P X (k) = e ikx = a x 2 + a 2. (4) e ikx P X (x)dx. (5) And P X (k) is also called the Fourier transform of P X (x). From normalization condition we have P X (k = 0) = 1, (6) and also P X (k) 1. An important PDF is the Gaussian P X (x) = ( ) 1 exp x2, (7) 2πa 2 2a 2 and in this case X is called a Gaussian random variable. The characteristic function of X is and its small k expansion, or long wave length behavior, is ( P X (k) = exp a2 k 2 ), (8) 2 P X (k) 1 a2 k (9) When X is a Lorentzian random variable, Eq. (4), its characteristic function is P X (k) = e a k, (10) its small k expansion is P X = 1 a k. (11) Unlike the Gaussian case the small k expansion for the Lorentzian PDF is non-analytical. This is related to the observation that the Gaussian PDF has a finite variance, while the

3 3 Lorentzian s variance diverges. More generally, the Taylor expansion of a characteristic function (ikx) P n X (k) = P X (x)dx = n=0 n! n=0 i n xn k n n! (12) provided that x n is finite, and i = 1. From Eq. (12) we see that x = i d dk P X (k) k=0, d 2 x 2 = 1 2 dk P 2 X (k) k=0 (13) etc, hence the characteristic function P X (k) serves as a moment generating function. For the Lorentzian PDF x 2 =, indeed d 2 exp( a k )/dk 2 k=0 =, and the analytical Taylor expansion in Eq. (12) is not valid. 1.1 Derive Eqs. (8) and (10). 1.2 Obtain the moments of the Gaussian PDF centered on x Show that exp( k µ ) with µ > 2 is not a characteristic function of a non-negative PDF. 1.4 More generally, show that if P X (k) 1 ik x a k µ, for k 0, then 0 < µ < Let X and Y be independent random variables with PDFs P X (x) and P Y (y). Find the PDF of Z = X/Y and Z = X + Y in terms of integrals of P X (x) and P Y (y). 1.6 Obtain the characteristic function of X, when P X (x) = 2π/(1 + x 2 ) 2. Consider now the random variable 0 < T <, whose PDF is P T (t). Such PDFs are called one sided. The Laplace transform of P T (t) ˆP T (u) = 0 e ut P T (t) dt, (14)

4 4 is also a moment generating function, since ( ut) ˆP n T (u) = P T (t) dt = 0 n=0 n! n=0 ( u) n T n. (15) n! Hence T n = ( 1) n d n du n ˆP T (u) u=0. (16) For the analytical expressions in Eqs. (15,16) to be valid, we require that moments of P T (t) are finite. An example were the integer moments of P T (t) diverge is the Smirnov PDF whose Laplace transform is P T (t) = 1 ( 2 π t 3/2 exp 1 ), (17) 4t ˆP T (u) = e u1/2. (18) Similar to the Fourier transform of the Cauchy PDF, the Laplace transform of Smirnov s PDF exhibits a non-analytical behavior in the vicinity of u 0 which has to do with the divergence of the first moment of the PDF. 1.7 Prove that Eqs. (17) and (18) are Laplace pairs. 1.8 Show that the small u expansion ˆP T (u) = A Bu α + O(u α ), must satisfy A = 1 B > 0 and α Construct a one sided PDF P T (t) whose n th moment diverges, while integer moments of order k < n are finite. Consider a generic one sided PDF which for long t satisfies P T (t) At (1+α) (19)

5 5 and 0 < A, 0 < α < 1. In this case P T (t) is moment-less e.g. t =. We wish to investigate the small u behavior of the Laplace transform ˆP T (u) = t0 0 P T (t)e ut dt + A t (1+α) dt t 0 ( +A t (1+α) e ut 1 ) dt + ɛ. (20) t 0 Here t 0 is large meaning that the approximation in Eq. (19) is valid for t > t 0. ɛ is small when t 0 is large. If P T (t) = At (1+α) for t > t 0 then ɛ = 0 and since we may choose t 0 as being as large as we wish we will take ɛ = 0. From the normalization condition the first two terms on the right hand side (RHS) of Eq. (20) are equal one when u 0. Also note that the small u expansion of the first term on the RHS of Eq. (9) is of the form C + Bu and as we show now the u dependence of this term is not a leading order term in the asymptotic expansion in the limit of u 0. Hence to investigate the small u behavior of P T (t) we must consider the integral ( A t (1+α) e ut 1 ) ( dt = Au α x (1+α) e x 1 ) dx (21) t 0 ut 0 We consider the limit ( lim x (1+α) e x 1 ) dx = Γ ( α), (22) ut 0 0 ut 0 where Γ() is the Gamma function. Collecting all the terms we find that if Eq. (19) holds then ˆP T (u) 1 + Γ ( α) Au α +, (23) when u 0. Such relations between small u behavior and large t behavior of Laplace pairs are called Tauberian and Abelian relations, they are discussed in the mathematical literature with rigor 2,3. Similar type of relations exists between small k behavior and large x behavior of Fourier pairs, as stated in the following question.

6 Consider the symmetric PDF P X (x) = P X ( x). Show that if P X (x) with 0 < µ < 2 then the characteristic function A x 1+µ x (24) P X (k) 1 B k µ k 0. (25) Obtain B in terms of A and µ Verify that rules Eqs. (23) and (25) hold for the examples of Smirnov s and Lorentz s PDFs respectively. III. GAUSSIAN CENTRAL LIMIT THEOREM 2,4 Let x 1,, x N be a set of N independent identically distributed random variables with a common PDF P X (x) whose mean is zero and variance is σ <. Consider the scaled sum Z = Ni=1 x i N 1/2. (26) According to the Central Limit Theorem, in the limit N P Z (z) 1 ( ) exp z2. (27) 2πσ 2 2σ 2 This is a remarkable theorem, of central importance in probability theory, statistical mechanics and beyond, since the PDF of the x i s is not necessarily a Gaussian still the scaled sum Z is Gaussian. Thus we say that the PDF P X (x) belongs to the domain of attraction of the Gaussian, if σ is finite. A poor man s proof of the Gaussian Central Limit Theorem is based on Fourier transforms. Let P X (k) be the characteristic function of x i. Since we assume x i = 0 we have

7 7 the small k behavior P X (k) 1 σ2 k A k γ (28) and 2 < γ. Using problem 1.5 the distribution of a sum of N independent random variable is a convolution, namely in Fourier space Hence using Eq. (28) we find P Z (k) = Taking the limit N we obtain [ ( )] N k P Z (k) = P X. (29) N 1/2 ( 1 σ2 2 k 2 ) N N + Akγ N. (30) γ/2 P Z (k) e σ2 k 2 2. (31) The inverse Fourier transform of which yields the Central Limit Theorem, Eq. (27). Two technical remarks are: (i) the choice of the N 1/2 scaling we introduced in Eq. (26) is justified in Eq. (30), choosing a different type of scaling e.g. Z = x i /N β with β 1/2 will not yield a meaningful limit theorem. (ii) The k γ term in Eq. (30) does not contribute to the N limit, hence information on behavior of P X (x), besides its variance is not important in the N limit. There exists a vast literature on the Gaussian central limit theorem, and more refined proofs and formulations of it yield insight into questions like how fast is the convergence of distribution of Z towards the Gaussian (i.e., what is the exact meaning of large N). An answer to this question is given in the Berry Eséen theorem, soon to be stated. Another question is where on the z axis, is the exact PDF P Z (z) well approximate by the Gaussian PDF. (i.e. when N is large but finite). In next chapter we will check this issue in the context of random walk theory, and we will show that under certain conditions that P Z (z) will behave like a Gaussian in its central part (hence the name Central Limit Theorem).

8 8 A. Berry Eséen Theorem 2 We consider the sum of N random variables Eq. (26), however for convenience use a dimensionless form by defining ξ = Z N /σ. Denote the PDF of ξ with φ N (ξ). The cumulative distribution function of ξ is Φ N (ξ) = ξ φ N (ξ)dξ. (32) We still assume x i = 0, x 2 i = σ2, and additionally the third moment r 3 = x i 3 < is assumed finite, and is determined of-course by the precise shape of P X (x). The Berry Eséen theorem states that Φ N (ξ) 1 2π ξ e ξ2 3r 3 2 dξ. (33) N 1/2 (σ 2 ) 3/2 This is a very strong bound on Φ N (ξ) since it does not depend on number of summands N. In the limit N we regain the Gaussian Central Limit Theorem. The bound yields an estimate on how far is the exact solution Φ N (ξ) from the asymptotic Gaussian behavior when N is finite. IV. LÉVY S GENERALIZED CENTRAL LIMIT THEOREM Let {x 1, x N } be a set of N independent identically distributed random variables, symmetrically distributed, with zero mean x i = 0, and with diverging variance x 2 i =. An example are random variables distributed according to the Lorentzian PDF Eq. (4). Clearly for this case the Gaussian Central Limit Theorem does not hold. Still as we show now an important generalization of the Gaussian Central Limit Theorem is valid. Let Z n = Ni=1 x i N 1/γ (34)

9 9 and the scaling exponent γ will soon be determined. The PDF of the x i is assumed of the form P xi (x) A x (1+µ) when x (35) and 0 < µ < 2 hence σ 2 =. In Fourier space P xi (k) 1 à k µ when k 0 (36) The A and the à are related, see question Eq below. Using an expansion very similar to that in Eq. (30) we have the characteristic function of the sum Z P Z (k) = ( ) N 1 à k µ N. (37) γ/µ where are terms of order higher than k µ. We now determine the scaling exponent, and choose γ = µ, (38) and find using Eq. (37) in the limit of large N P Z (k) e à k µ. (39) The characteristic function of Z has a form of a stretched exponential. Inverse Fourier transform of Eq. (39) yields the PDF of Z in the limit of N, such PDFs are called stable PDFs, or sometimes Lévy s PDFs and are denoted here with l µ, Ã,0 (z). The PDF l µ, Ã,0 (z) is symmetric which is obviously related to the fact that we assumed that the x i s is symmetrically distributed random variable. The expenent µ is called the characteristic exponent, and à is a scale factor. In the special case µ = 2 we recover from Eq. (39) the Gaussian PDF. For µ = 1 we obtain the Lorentzian PDF. For other choices of µ Schneider showed that l µ, Ã,0 (z) can be expressed in terms of Fox functions, though generally they are

10 10 not tabulated. Hopefully this will change in the near future, and programs like Mathematica will yield stable PDFs.. Obtaining stable PDFs is in principle easy, using numerical inverse Fourier transform. We investigate some of the Mathematical properties of l µ, Ã,0 (z) in the Appendix Show that µπ A = Ãsin Γ (µ) µ 2. (40) π 1.13 van-kampen writes The unfortunate name stable is used for distributions having the property that the sum of two variables so distributed has again the same distribution (possibly shifted and/or rescaled). Show that the symmetric Lévy PDFs Eq. (39) are stable. Is the term stable PDFs unfortunate or is van-kampen s unfortunate remark unfortunate. V. LAWS OF LARGE NUMBER S Consider the sum of N random independent identically distributed random variables t N = Ni=1 t i N β, (41) and t i > 0 have a common PDF P t (t). Let ˆP t (u) be the Laplace transform of P t (t). When β = 1, t N is a random variable, however we expect that in the limit lim N β=1 = t = N 0 tp t (t)dt, (42) provided that t is finite. The law of large numbers considers the mathematical meaning of the identity in (42). Here we will consider this law briefly, using Laplace transforms ˆP tn (u) = [ ( )] u N ˆPt. (43) N β

11 11 Using the small u expansion we have two classes of PDFs, those with infinite t ˆP t (u) = 1 Au α + 0 < α < 1, (44) and those with a finite first moment ˆP t (u) = 1 t u + u 0. (45) Using Eq. (43,45) we have for the case when the underlying PDF of {t i } is moment-less we choose β = 1/α and find ˆP tn (u) = (1 Auα + )N, (46) N αβ lim N ˆP tn (u) = e Auα. (47) For the second class of PDFs, those with finite first moments Eq. (44), we obtain lim N ˆP tn (u) = e t u. (48) and hence as expected For α < 1 we find that lim P t N N (t) = δ(t N t ). (49) lim N P t N (t) = l α,a,1 (t). (50) l α,a,1 (t) is the one sided Lévy stable PDF which is the inverse Laplace transform of l α,a,1 (u) = exp( Au α ). We see that when α < 1 the average t N remains random even when N. Some mathematical properties of one sided Lévy PDFs are discusses in Appendix A. VI. GENERATING LÉVY FLIGHTS ON A COMPUTER Further insights into power law statistics is gained through simple numerical experiments. It is easy to generate on a computer a random number, call it x, uniformly distributed in

12 12 the interval [0, 1]. Starting with such a random number we seek a transformation which will yield a random variable which is described by power law statistics. One such transformation is t = (1 x) 1/α. (51) Using the chain rule dx P t (t) = P x (x) dt, (52) and the uniform PDF P x (x) = 1 when 0 < x < 1 we find P t (t) = αt (1+α) (53) and 1 < t <. This is only one example of how we can generate a random number whose PDF decays like t (1+α) for long times. In some cases we may wish to obtain random variables t described by a specific PDF P t (t) and for which simple transformations like Eq. (51) are difficult to obtain. In that case a method called the Accept-Reject method can be used, the latter is discussed in 1. Figures: Devil Staircase, Exp Stair case, Lévy Flight Invent a transformation that maps a uniformly distributed random variable, onto a power law random variable whose PDF behaves like P t (t) t (1+α), and which is different than Eq. (51) Consider the sum T N = N i=1 t i and t max = maximum of {t i }. Obtain T N and t max M times. Plot T N versus t max for two cases: (i) t i uniformly distributed between [0, 1] and (ii) for a one sided power law PDF with α < 1 e.g. use Eq. (51). Use large M and N and explain your observations.

13 Consider a sum of N random variables uniformly distributed in [ 1/2, 1/2], use numerical simulation to observe Gaussian Central Limit behavior. For what values of N do you expect a reasonable convergence to Gaussian behavior. Note that you may also obtain exact solution to the problem in Fourier space, and then invert the solution numerically Consider a sum of N random variables X, with P x (X) x 2 when X, and P X (x) = P X ( x). Use numerical simulations to observe Lévy Central Limit behavior. VII. LONG RANGE INTERACTIONS AND LÉVY STATISTICS Consider a system of particles interacting via a two body central field. Select one of the particles called the tracer particle, and place it on the origin. The density of particles is ρ = N/V, where N is the number of particles and V is the volume. In the thermodynamic limit both N and V are large, while ρ is finite. The force the particles exert on the test particle is N F = F( r i ). (54) i=1 For example for a point masses m embedded in a sea of other point masses M F( r i ) = GmM ˆr r 3 (55) and all other symbols have their usual meaning. In some physical situations the particles are uniformly distributed in space. Then we may think of an ensemble of tracer particles, each embedded in a sea of bath particles, and ask what is the distribution of forces on the tracer particle. The problem is a problem of summation of random variables. This type of problem appears in many brunches of physics, with long range interaction. Chandrasekar considered it in the context of the calculation of the distribution of forces

14 14 acting on a gravitational object. Stoneham s theory of inhomogeneous line broadening of defected crystals, can be interpreted in terms of Lévy statistics. Klauder and Anderson 18 showed Lévy statistics is important in the context of spectral diffusion in spin systems governed by dipolar interaction (note that these authors do not use the term Lévy statistics to describe their results). Such Lévy statistics describe statistics of single molecule spectroscopy in low temperature glasses ( 14 for theory, and 15 for experiment) and also certain aspects of turbulence. It was Holtsmark who considered this problem first in the context of spectroscopy. Let us consider a specific example and let us calculate the distribution of forces along the z direction for the Newtonian attraction in three dimensions F z = N i=1 GmM cos(θ i ) r i 2. (56) The factor cos(θ i ) yields the projection of the force on the z axis. We now consider the distribution of F z, assuming a uniform distribution of the other objects (a rather strong assumption, which we discuss later). We find the characteristic function of F z, making use of the fact that the bath of particles are statistically uncorrelated (Physically we neglect the interaction of bath particles) we find [ e ikfz V = 1 + dv ] (eigmm cos(θ)/ r 2 1) N (57) V Here I used the fact that a bath star is uniformly distributed in space, hence its density is 1/V. In the thermodynamic limit N we have [ ] e ikfz = exp ρ dv (1 e ikgmm cos(θ)/ r 2 ). (58) V The integration of the imaginary part in the exp, vanishes because of symmetry V dv sin[kgmm cos(θ)/ r 2 ] = 0,

15 15 hint using spherical coordinates 1 1 sin(cos(θ))d cos(θ) = 0. Hence the integral to solve is { e ikfz = exp ρ dv [ 1 cos(gmm cos(θ)/ r 2 ) ]}. (59) V The fact that the integral is real means that the PDF of F z is symmetric, and we are equally likely to find F z > 0 or F z < 0. We can also consider only only k > 0 since e ikfz must be an even function of k. After a change of variables and integration over angles [ e ikfz = exp 8πρ ] 5 (GmM)3/2 k 3/2 dyy 2 (1 cos 1/y 2 ). (60) 0 The integral is solved according to 0 dyy 2 (1 cos 1/y 2 ) = 2π 3. Eq. (60) shows that the PDF of F z is a symmetric Lévy stable law, with a characteristic exponent 3/2 (because of the k 3/2 ). The variance of the force in z direction is infinite. The distribution was found by Holtsmark in a different context, a long time ago. Why did we get Lévy statistics for this problem. Since we assumed that particles are uniformly distributed in space, and independent of each other the problem is similar to the mathematical problem of summation of independent random variables. Some of the terms in the sum Eq. (56) may become very big, when two gravitational objects i.e. stars, are closely situated. On the other hand many other terms are typically very small, due to the stars in the background. This causes a wide distribution of forces and leads to Lévy statistics. We assumed that the system of point masses is uniformly distributed in space, while Chandrasekar seems to claim that this is a reasonable assumption it clearly must be doubted. Interaction among the bath particles will lead to complex correlations among their position in space. However, in other disordered condensed matter systems uniformly distributed defects leading Lévy statistics are indeed quite common. For example low density quenched

16 16 defects in crystals, for example dislocations, are to an excellent approximation randomly distributed in space. And these defects when interacting with an optical marker, create the so called inhomogeneous line broadening effect. The physics is as follows. An atom in a gas phase has a certain absorption frequency ω 0. This atom is embedded in a crystal, and it interacts with the randomly and uniformly distributed defects. These defects will cause a frequency shift, namely the atom s absorption frequency is ω 0 + ω i. The ω i are due to interactions of the atom with defects, they are usually calculated using quantum mechanics in particular first order perturbation theory 13. The details of these interaction are an issue for a condensed matter course. Here we are interested only in certain statistical aspects of the problem, and we claim without proof that in many cases, the frequency shifts decay like ω i r δ i where r i is the distance of the defect to the optical marker. δ will depend on details of the interaction of the atom and defect, however for simple interaction like Coloumb interactions, interactions with certain strain defects, or dipoles one finds δ = 1, or δ = 2, or δ = 3. The important point is that the defects are uniformly distributed in space with a density ρ, and hence: 1.18 Show that the PDF of ω = ω 0 + N i=1 A cos θ i r i δ (61) in d dimensions is a symmetric Lévy stable law with a characteristic exponent d/δ. Show that Holtsmark PDF is a special case. Explain why the choice of angular dependence (i.e. the artificial cos θ term) does not change your result in qualititative way (universality) provided that on average negative and positive frequency shifts are equally likely (symmetry).

17 17 Since the crystal will have in it many markers (called chromophores), these optical centers will have a very wide Lévy spectrum of frequencies. Namely the absorption line shape of a single atom in gas phase is very narrow, the width being the inverse natural life time of the excited state. In contrast the absorption line of an ensemble of such atoms embedded in a crystal, is in many cases very broad its shape being a symmetric Lévy function. The particular type of L evy function will depend on the type of defects in the sample (see Table 1 for some details). This topic is very nicely reviewed by Stoneham, who unfortunately did not recognize the relation of this problem with Lévy statistics. Table Summary of Stoneham s theory on inhomogeneous line shape broadening, for different types of interactions, and dimensions d. The results show that normalized inhomogeneous lines are symmetrical stable laws l d/δ,0 (ω). L 10 is the Lorentzian, L 3/2,0 is Holtsmarks function, and L 2,0 (ω) is a Gaussian with logarithmic corrections.

18 18 Physical problem d δ Line Shape Strain Broadening due to dislocations 2 1 L 2,0 (ω) Strain due to dislocation dipoles 2 2 L 1,0 (ω) Strain broadening due to point defects 3 3 L 1,0 (ω) Magnetic and electric dipole broadening 3 3 L 1,0 (ω) Broadening due to random electric fields 3 2 L 3/2,0 (ω) An important issue are cutoffs. We derived the Lévy distribution assuming that the point defects (or masses) are uniformly distributed in space. However Physically two objects cannot come into close contact with each other, for example a star has a finite size. Or the 1/r δ interaction is not valid for short distances since particles repel each other on short distances. We also assumed that the size of our system is infinite, while in practice it is finite. These lower and upper bounds can be included in the theory and then non-universal deviations from Lévy behavior can be observed. These deviations depend ofcourse on the details of the problem, for example what exactly are the cutoffs etc. Without going into details if δ >> d usually a small cutoff is sufficient to destroy the Lévy behavior. Thus in Physics we expect Lévy behavior only when the interactions are long range, namely δ d. The reason is clear only if the interaction is long ranged, then we have a summation of many

19 19 random variables (i.e. the forces or frequency shifts), and a possibility for a central limit theorem argument to hold. 1 Press Numerical recipes 2 Feller 3 G. Weiss 4 N. G. van-kampen Stochastic Processes in Physics and Chemistry (North Holland Amsterdam 1981). 5 Abramovitz 6 G. Samorodnitsky and M. S. Taqqu Stable Non-Gaussian Random Processes (Chapman and Hall/CRC New York 2000). 7 B. V. Gnedenko, and A. N. Kolmogorov Limit distributions for sums of independent random variables (Addison-Wesley, Reading MA 1954). 8 W. R. Schneider in Stochastic Processes in Classical and Quantum Systems edited by S. Albeverio, G. Casatti, and D. Merlini (Springer, Berlin 1986). 9 H. Scher, and E. Montroll Phys. Rev. B (1975). 10 E. Barkai, Phys. Rev. E (2001). 11 E. W. Montroll, and J. T. Bendler, J. Stat. Phys (1984). 12 E. W. Montroll, and B. J. West, On an Enriched Collection of Stochastic Processes in Fluctuation Phenomena Chapter 2, North Holland Publishing Company (1979) Montroll and Lebowitz Editors. 13 A. M. Stoneham Rev. Mod. Phys (1969). 14 E. Barkai, R. Silbey, and G. Zumofen Phys. Rev. Lett (2000).

20 20 15 E. Barkai, A. V. Naumov, Yu. G. Vainer, M. Bauer, and L. Kador Phys. Rev. Lett (2003). 16 Holtsmark 17 I. A. Min, I. Mezik, and A. Leonard Phys. Fluid (1996). 18 J. R. Klauder and P. W. Anderson Phys. Rev (1962). VIII. APPENDIX A: MATHEMATICAL PROPERTIES OF STABLE PDFS A. Symmetric Stable Laws We have found that an important class of stable PDFs are symmetric stable laws whose Fourier transform is exp( A k µ ). The inversion of this characteristic function is now considered. Besides the case µ = 1 (Lorentzian) and µ = 2 (Gaussian) the inversion is not straight forward. Schneider 8 expresses the symmetric Lévy stable laws in terms of certain Fox functions. These Fox functions are generally not tabulated, though one can use known asymtotic behavior of Fox functions to derive asymptotic behaviors of Lévy stable laws. We will consider the case A = 1 since the generalization to A 1 is straight forward. Consider the symmetric Lévy PDF l 1,µ,0 (x) = 1 e ikx k µ dk = 1 cos kxe kµ dk. (62) 2π π 0 Integrating by parts and changing the integration variable we have l 1,µ,0 (x) = µ k µ 1 sin kxe kµ dk = µ e ( x) v µ v µ 1 sin vdv. (63) πx 0 πx µ+1 0 For large x l 1,µ,0 (x) µ v µ 1 sin vdv (64) πx µ+1 0

21 21 and hence if 0 < µ < 1 then l 1,µ,0 (x) µ µπ Γ(µ) sin πx1+µ 2. (65) It is easy to see that this large x behavior is valid also for the Lorentzian case µ = 1. If 1 < µ the integral Eq. (64) is not well defined and a different approach must be used. The proof that Eq. (65) is still valid for 1 < µ < 2 is slightly more complicated l µ,1,0 (x) = 1 π change variable ω = kx and let η = x µ and then For large x hence small η 0 cos(kx)e kµ dk l µ,1,0 (x) = 1 e ηω (cos ω)e η(ωµ ω) dω. πx 0 l µ,1,0 (x) 1 e ηω cos ω [1 η (ω µ ω) + ] dω. (66) πx 0 The integrals are solved using a formula from a Table (e.g. Mathematica) 0 ω µ cos ωe ωη dω = ( 1 + η 2) (1+µ)/2 η 1 µ cos [(1 + µ)arctan(1/η)] Γ (1 + µ). Using Eq. (66) l µ,1,0 (x) 1 { η πx 1 + η + η η2 1 2 (1 + η 2 ) η(1 + 2 η 2 ) (1+µ)/2 η (1+µ) cos [(1 + µ)arctan( 1η ] } ) Γ(1 + µ). In the limit of x corresponding to η 0 we obtain Eq. (65). B. One Sided Stable Laws One sided Lévy stable laws, l α,1,1 (t) 0 < t < defined via their Laplace transform ˆlα,1,1 (s) = exp( s α ), are now investigated. Generalization to the case l α,a,1 (t) with A 1 is straightforward. Recall also that 0 < α < 1.

22 22 We write the stable density in terms of Laplace inversion integral a change of variables st = z yields l α,1,1 (t) = 1 c+i dse st e sα, (67) 2πi c i tl α,1,1 (t) = 1 c+i dze z ( z/t)α. (68) 2πi c i We consider 1/t α << 1, hence expand exp( z α /t α ) in Taylor series tl α,1,1 (t) = 1 1 2πi n! n=0 ( 1 ) n dz( z) nα e z. (69) t α C The integral in this equation is the Hankel representation of the Γ function 5. Thus we have l α,1,1 (t) = n=0 Using the reflection formula for Γ functions 5 1 n! ( 1) n 1/Γ ( αn). (70) tnα+1 Γ (z) Γ (1 z) = π csc πz, (71) we find l α,1,1 (t) = 1 π n=1 Γ(1 + nα) n! ( 1) n 1 sin(πnα)t (αn+1). (72) The opposite limit of small t can be treated using steepest descent method (Scher-Montroll). A summary of some properties of one sided stable PDFs is given in Appendix A of 10.