Integrated random processes exhibiting long tails, finite moments and 1/f spectra

A dynamical model based on a continuous addition of colored shot noises is presented. The resulting process is colored and non-Gaussian. A general expression for the characteristic function of the process is obtained, which, after a scaling assumption, takes on a form that is the basis of the results derived in the rest of the paper. One of these is an expansion for the cumulants, which are all finite, subject to mild conditions on the functions defining the process. This is in contrast with the Levy distribution -which can be obtained from our model in certain limits- which has no finite moments. The evaluation of the power spectrum and the form of the probability density function in the tails of the distribution shows that the model exhibits a 1/f spectrum and long tails in a natural way. A careful analysis of the characteristic function shows that it may be separated into a part representing a Levy processes together with another part representing the deviation of our model from the Levy process. This allows our process to be viewed as a generalization of the Levy process which has finite moments.

property on the probability distribution and obtain several relevant functions. Section V is devoted to moments and cumulants and the evaluation of the power spectrum which shows the 1/f character of our process. In Section VI we study the asymptotic behavior of the distribution while in Sect. VII we present, starting form our distribution, an Edgeworth-type series for the Lévy distribution. Conclusions are drawn in Sect. VIII and some more technical details are in Appendices.

II. MAIN RESULTS
In this section we wish to state the main results of the paper without giving the derivations or being careful to state the range of validity for which they hold. We hope that this will give a good indication of the scope and nature of our results; the interested reader can then fill out this basic framework by proceeding to later sections.
As explained in the next section, our integrated process X(t) is a continuos superposition of colored shot noises parametrized by u. It takes the form There are several components: (i) The pulse shape function, φ(t, u), (ii) the jump amplitude A k (u) of the kth pulse, (iii) the jump time T k (u) of the kth pulse. The jump amplitudes, A k (u), are identically distributed random variables distributed according the probability density function (pdf) h(x, u), i.e. h(x, u)dx = Prob{x < A k (u) < x + dx}. The jump times are assumed to follow a Poisson process with parameter λ(u). Therefore the model is characterized by the functions φ(t, u), h(x, u) and λ(u). There are very few restrictions on these functions (one is the causality condition φ(t, u) = 0 for t < 0) but we will argue that it is natural to assume the scaling forms where σ(u) is the standard deviation of the jump amplitudes. The model is now specified by the four functions of a single variable φ, h, σ and λ.
One of the main goals of this paper is to find the probability distribution of the process X(t), this distribution will be obtained through the characteristic function (cf)p(ω, t) = exp[iωX(t)] , which is the Fourier transform of the density p(x, t). By assuming the scaling forms given above the one time distribution of X(t) is explicitly given by the following cfp (ω, t) = exp −btω α Contrary to Lévy processes, our integrated process can have finite moments of any order, thus cumulants are given by x n/α dx, (n = 1, 2, 3, · · ·). Note that the second cumulant, (i.e., the variance), is proportional to (bt) 2/α and X(t) presents anomalous diffusion behavior (see Sect. V for a detailed discussion on limiting values and bounds for exponent α and other parameters related to the asymptotic behavior of the pulse shape function φ(x)).
We define the stationary correlation function by C(τ ) = lim t→∞ X(t + τ )X(t) . For our process this reads The power spectral density of the process X(t), given by the Fourier transform of the stationary correlation, is and X(t) is 1/f noise with exponent ν = 1 + 2/α.
We also can perform the asymptotic analysis of the probability distribution of X(t) without having to specify any particular form for h(x) and φ(x) thus keeping the maximum level of generality. Specifically we show in Sect. VI that the center of the distribution is approached by a Lévy distributioñ where 0 < δ < 2. We refer the reader to see Sect. VI for more details and for the behavior of the tails of the distribution which are mainly determined by the behavior of the jump pdf h(x). The relation to the Lévy distribution is explored in more detail in Sect. VII where we present and alternative (and exact) expression for the cf which decompose the distribution of the integrated process X(t) into that of Lévy plus an additional term: Note that when φ(x) is the rectangular step function this equation reduces to the Lévy distribution. Therefore, when φ(x) is a step-like function close to the Heaviside function, this alternative expression can be used as the starting point of an Edgeworth-type expansion procedure giving corrections to the Lévy distribution.

III. THE INTEGRATED PROCESS
Let X(t) be a random process formed by a continuos superposition of independent shot-noise processes: where for any fixed time t, Y (u, t) are independent random variables for different values of parameter u (see Eq. (10) below) and for any fixed value of u, Y (u, t) is a colored shot-noise process represented by a countable superposition of pulses of identical shape: where T k (u) marks the onset of the kth pulse, and A k (u) is its amplitude. Both T k (u) and A k (u) are independent and identically distributed random variables with probability density functions given by h(a, u) and ψ(t, u), respectively. The pulse shape φ(t, u) has to fulfill the "causality condition", i.e., φ(t, u) = 0 for t < 0 [16]. We assume that the occurrence of jumps is a Poisson process, in this case the shot-noise Y (t, u) is Markovian, and the pdf for the time interval between jumps, ψ(t, u)dt = P{t < T k (u) − T k−1 (u) < t + dt}, is exponential: where λ(u) is the mean jump frequency, i.e., 1/λ(u) is the mean time between two consecutive jumps [17]. We recall that jump amplitudes A k (u) are identically distributed (for all k = 1, 2, 3, · · ·) and independent random variables (for all k and u). In what follows we will assume that they have zero mean and a pdf, h(x, u)dx = P{x < A k (u) < x + dx}, depending on a single "dimensional" parameter which, without loss of generality, we assume to be the standard deviation of jumps σ(u) = A 2 k (u) . That is, Before proceeding further with the probability distribution of the integrated process X(t) given by Eq. (1), we note that following Rice's method [16] one can easily obtain all the probability distributions of the shot noise Y (t, u) via their cf'sp Y (ω 1 , t 1 ; · · · ; ω n , t n ; u) = exp i n k=1 ω k Y (t k , u) .
In Appendix A we show that and (supposing that t 2 > t 1 ) whereh(ω) is the Fourier transform of the jump pdf h(x).
Let us now evaluate the probability distribution of the integrated process X(t). In terms of the cumulants, Y (t, u) of the shot noise Y (t, u) we see that the one time characteristic function of X(t) can be written as That is, But, by our assumptions on the process Y (u, t) we have Therefore, Note that this line of reasoning can be easily extend to the nth time distribution, with the result: Going back to our integrated process we have from Eqs. (6)-(7) and (11)-(12) that the one time characteristic function of X(t) reads while the two time cf is (t 2 > t 1 ) where we have dropped the subscript X. Obviously these are formal expressions, as long as we do not provide the functional dependence of λ(u) and σ(u) on the parameter u. We will do so in the next section using scaling arguments.

IV. SCALING
In order to proceed further we need to specify the functional form of λ(u) and σ(u). Of course that form will depend on the specific features of the problem at hand. At this point we choose what seems to us one of the most general ways of proceeding, we thus suppose that our integrated process X(t) possesses self-scaling properties. Following this path we must first assume that the pulse function is of the form which turns φ(u, t) into a function of the single dimensionless variable λ(u)t. Substituting this into Eq. (13), defining new integration variables s = t ′ /t and z = ωσ(u), and supposing that σ(−∞) = 0 and σ(∞) = ∞, we obtain where the prime on σ denotes derivative. We now impose the self-scaling property on the cf, that is, we assume that p(ω, t) is a function of the single variable ωt 1/α :p On the other hand we note that in Eq. (16), the quantities σ ′ and λ are functions of z and ω. Then scaling (17) implies where A(z) and B(z) are arbitrary functions to be determined. From these two relations we get σ ′ = C(z)/ω, where C(z) = B(z)/A(z). In the original variable u we have But σ ′ = σ ′ (u) is independent of ω. Therefore, the unknown function C has to be of the form C(ωσ) = kωσ where k is a constant. Hence σ ′ (u) = kσ(u), whence σ(u) = σ 0 e ku . Finally absorbing the constant k inside the variable u we obtain the functional dependence of the jump variance σ on parameter u: where σ 0 is a constant. Moreover we see from Eq.
is an arbitrary constant. Substituting this into the first relation of Eq. (18) yields the "dispersion relation" between the mean frequency λ and the jump variance σ: The functional dependence of λ on u is obtained by combining Eqs. (20)-(21), Collecting results we see from Eq. (16) and Eqs. (20)-(23) that the one-time characteristic function reads It is sometimes convenient to rewrite this equation and use the alternative form ofp(ω, t) given bỹ or equivalentlyp Starting from Eq. (14) and following an analogous reasoning based on the scaling assumption, we obtain the following expression for the two time characteristic function of the integrated process (with t 2 > t 1 ) Equations (24)-(27) are some of the key results of the paper. Since, as we will see next, they constitute a generalization of the Lévy distribution with finite moments.

V. MOMENTS, CUMULANTS AND POWER SPECTRUM
We first note from Eq. (24) that if the pulse shape function is the Heaviside step function: then the integrated process X(t) is identically a Lévy process, regardless the jump pdf h(x): where Therefore, following our model, Lévy processes can be viewed as a continuos superposition of families of rectangular pulses occurring at random Poisson times. The usual range of the exponent α in Lévy flights is 0 < α < 2. In such a case X(t) has no finite moment but the first one [18]. In actual situations, one is unlikely to meet with perfect rectangular pulses (showing sudden changes) in such a case all moments can be finite and are easily evaluated from Eq. (25) through Thus for instance the second moment is given by (note that due to Eq. (4)h ′′ (0) = −1) As an illustrative example suppose that our pulse function has the form where k > 0 is a constant (note that if k is large then φ(t) approaches to the rectangular pulse (28)). In the Appendix B we show that (see also Eq. (43) below) where D = αb 2 k −1+2/α (1 − 2 2/α−2 )Γ(2 − 2/α)/(2 − α). Since 1 < (2/α) < 3, Eq. (33) clearly shows a superdiffusive behavior.
In fact we can easily obtain a closed expression not for the nth moment but for the nth cumulant ¿From Eq. (25) we have If in the double integral on the right hand side of this equation we define a new integration variable x by s = (z α /bt)x and exchange the order of integration we get but the last integral is trivially evaluated, and for the nth cumulant we have Taking into account thath (n) (0) = 0 for n odd (we have assumed a symmetric jump distribution h(x)) we write and (n = 1, 2, 3, · · ·). In order to check the convergence of these expressions and the existence of moments, we first assume that (β > 0) then the convergence of the integral on the right hand side of Eq. (36) as x → 0 implies that the scaling exponent α has a lower bound: On the other hand if we assume that then the convergence of (36) when x → ∞ implies that the scaling exponent α also has an upper bound: Moreover when γ ≥ 0 then if all cumulants will exist. Note that Eq. (40) holds whenever γ ≥ −1/2n. Therefore, γ ≥ 0 is a sufficient condition for its validity. On the other hand if γ < −1/2 there is no upper bound on the accepted values of α. We also observe that for a step-like function, as that of Eq. (32), where γ = 0 then all cumulants will exist if 2 > α > 1/β.
Finally, for any integrable function φ(t) over [0, ∞) there is no upper bound for α and the only condition on α for having all moments finite is that α > 1/β. We close this discussion on moments and cumulants with an example. Suppose that the pulse function is given by the step-like function (32). In this case γ = 0, β = 1 and all moments (and cumulants) will exist if 1 < α < 2. Cumulants are given by Eqs. (35)-(36). In Appendix B we show that where the numbers A n are given by Eq. (B3) of Appendix B. Finally, We finish this section evaluating the power spectrum of the integrated process X(t). Let us first evaluate the correlation function ¿From Eq. (27) we get Let C(τ ) be the correlation function in the stationary limit t → ∞, i.e.
¿From Eq. (44) we have Note that the (stationary) variance C(0) = ∞ which agrees with the superdiffusive behavior of X(t) given by Eq. (33). The power spectral density of our process is thus given by the Fourier transform of the stationary correlation functionC Substituting Eq. (45) into this equation, performing simple changes of variables and taking into account the causality of the pulse function φ(t) we finally obtainC where andφ(ξ) is the Fourier transform of φ(t). The power spectral densityC(ω) exists if the integral In order to prove the existence of J we first need that the Fourier transform of the pulse function,φ(ξ), exists. Note that any step or step-like function does not have a Fourier transform and consequently the power spectrum is infinite. For the existence ofφ(ξ) it suffices that φ(x) be absolutely integrable, and from the asymptotic behavior given by Eq. (39): we have to impose that γ < −1. This in turn implies thatφ(ξ) ∼ ξ −1−γ as ξ → 0 and, since γ < −1, the integral J at its lower limit is always finite for any α > 0. On the other hand if φ(x) satisfies Eq. (37) as x → 0, thenφ(ξ) ∼ ξ −1−β as ξ → ∞. Hence, J will be finite if α > 1/(β + 1/2) (see also Eq. (38)). Therefore, the process X(t) has a finite power spectrum if φ(x) is absolutely integrable on the real line, and X(t) has a finite second cumulant, Eq. (38), i.e.

VI. ASYMPTOTIC BEHAVIOR
We will now examine the asymptotic behavior of the one time probability density function (pdf) of the integrated process X(t), p(x, t)dx = P{x < X(t) < x + dx}. For this analysis we distinguish two regions: the "center" (x → 0) and the "tails" (x → ±∞) of the distribution. We cannot have a closed expression for the pdf p(x, t) until the pulse function φ(x) and the jump pdf h(x) are both specified. Therefore, we will perform the asymptotics on the cfp(ω, t). As a well known feature of the harmonic analysis the center of the distribution is determined by the large ω behavior of the cf, while the tails are determined byp(ω, t) when ω → 0 [19].
We deal first with the center of the distribution where ω → ∞. If we assume that the pulse function φ(x), as x → ∞, satisfies Eq. (39): Substituting this into Eq. (24) and performing the change of variables ξ = (btω α ) γ z 1−αγ we obtain the following Lévy distribution:p where Note that due to the bounds discussed above (see Eq. (40)) we have 1 − αγ > α/2 hence the Lévy exponent in Eq. (50) satisfies and Eq. (50) is well defined. We also note that for a step-like pulse function φ(t) where γ = 0 we obtain the same Lévy distribution, Eq. (29), that satisfies the model for sudden pulses (28). Therefore, for any pulse shape function satisfying condition (39) the center of the pdf is given by a Lévy distribution. Let us now obtain an asymptotic expression of the pdf p(x, t) when x → ±∞, which will be valid if φ(t) obeys Eqs. (37) and (39), and the exponent α is bounded by β −1 < α < 1/(γ + 1/2).
Substituting this into Eq. (54) we see that Therefore the tails of the distribution are determined by the jump pdf h(x) and the pulse shape function φ(t).
Finally for the rectangular pulse (28) we have which agrees with the expected tail behavior of the Lévy distribution [18].

VII. RELATION TO THE LÉVY DISTRIBUTION
In section V we obtained general expressions (35)-(34) for all of the cumulants, from which it follows that In addition, in section VI, we showed that p(x, t) was a Lévy distribution at the center of the distribution (x → 0) and took the form (55) in the tails (x → ±∞) of the distribution. In this section we will show how the distribution can be separated into a Lévy distribution plus an additional term. This term takes the form of a single integral which can be evaluated once the functions φ and h have been specified. We begin the analysis by changing variables from s to x = bts/z α (z fixed) in Eq. (25). This gives changing the order of integration. At this point we factor out the contribution from the Lévy process by writing (58) as or, after definingg as The first term is just lnp(ω, t) for the Lévy processes (see Eqs. (29)-(30)). The second term can be simplified by first writing it as and then integrating by parts to give We assume thath(ω) is analytic at ω = 0 and integrable, theñ and since 0 < α < 2 the first term in Eq. (60) is zero. Finally where φ ′ (x) is the derivative of the pulse shape function and lnp Levy (ω, t) = −M tω α where M is given by Eq. (30). Note that when φ(x) is the Heaviside step function the integrals on the right hand side of Eq. (61) vanish and Eq. (61) reduces to the Lévy distribution. Therefore, we can look at the second term on the right hand side of Eq. (61) as a correction to the Lévy distribution when φ(x) is not exactly a Heaviside function but a step-like function very close to the Heaviside function. This may be evaluated, in principle, for any given φ and h. For instance we could take φ to be of the form (32) with k large and the Lorentziañ The form of the correction terms depends on the choice of the functions to an extent, and so we will not discuss the explicit form it takes here.

VIII. CONCLUSIONS
In this paper we have presented and analyzed a dynamical model based on a process which is a superposition of colored Poisson noises. The model was shown to have several attractive features. The probability density function has long tails which emerged in a natural way and, unlike the Lévy distribution, all the moments of the distribution are finite. We believe that these properties make the distribution an ideal candidate for describing stock market prices [6].
A property what may have more relevance to physics and other natural sciences is the appearance of 1/f noise in the power spectrum. Once again we would like to stress that this result flowed naturally from the nature of the model and the scaling assumptions which reduce h and φ from functions of two variables to functions af a single variable.
In a more mathematical context, we believe that the decomposition of the cf of our model into that of the Lévy plus additional terms is interesting, both as an example of an Edgeworth-type expansion and for the nature of the corrections to the Lévy distributions when the parameters of our model are chosen so that our distribution is near to the Lévy one.
There are still some open questions. One of them is the extension of the model to the increments of the process , since in this case we believe that the process Z(τ, t − t 0 ) becomes stationary when it starts in the infinite past (t 0 → ∞). Another interesting and open question is the actual application of the model to financial time series where some non-white correlation is observed [20]. Both points are presently being investigated. By generalizing Rice's method [16], we will now obtain the probability distribution of the shot noise Y (u, t) defined by Eq. (2): where we assume that the random variables A k (u) and T k (u) are identically distributed and statistically independent. The jump amplitudes are described by the pdf h(x, u)dx = Prob{x < A k (u) < x + dx} and the jump times T k (u) follow a Poisson distribution of parameter λ(u). Define to be the joint pdf of the process with t 2 ≥ t 1 . This pdf can be written as p(x 1 , t 1 ; x 2 , t 2 ; u|n 1 , n 2 )P (n 1 , t 1 ; n 2 , t 2 ; u), where p(x 1 , t 1 ; x 2 , t 2 ; u|n 1 , n 2 ) is the conditional pdf assuming that exactly n 1 pulses have occurred at time t 1 and n 2 pulses at time t 2 . P (n 1 , t 1 ; n 2 , t 2 ; u) is the joint probability for the occurrence of such pulses. Since t 2 ≥ t 1 then n 2 ≥ n 1 and where is the Poisson distribution. Substituting Eq. (A3) into Eq. (A2) and defining t 1 = t, t 2 = t + ∆t, n 1 = n and n 2 − n 1 = m, we obtain p(x 1 , t; x 2 , t + ∆t; u|n, n + m)P (m, ∆t; u)P (n, t; u), and the characteristic function reads (ω 1 , t; ω 2 , t + ∆t; u|n, n + m)P (m, ∆t; u)P (n, t; u). (A5) Note thatp(ω 1 , t; ω 2 , t + ∆t; u|n, n + m) is the joint characteristic function of the truncated process