Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution ($\Delta$) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we show a direct comparison between high energy physics and quantitative finance results. Both for physics and finance, we illustrate that correlations between particles lead to a broadening of the multiplicity distribution and to dynamical fluctuations when the resolution becomes small enough. From the generating function of factorial moments, we make a prediction on the gap probability for sequences of returns of positive or negative signs. The gap is defined as the number of consecutive positive returns after a negative return, thus this is a gap in negative return. Inversely for a gap in positive return. Then, the gap probability is shown to be exponentially suppressed within the gap size. We confirm this prediction with data.
Multiplicity distributions and correlations between final-state particles in nuclear interactions are an important testing ground for analytic perturbative theory, as well as for Monte-Carlo (MC) models describing the hadronic final state [1]. Two-particle angular correlations have been extensively studied experimentally [2]. Specific statistical tools, namely the normalized factorial moments, have emerged in order to analyze in much details multiplicity distributions measured in restricted phase-space regions. The normalized factorial moments are defined as F q (∆) = n(n -1) . . . (n -q + 1) / n q , q = 2, 3, . . . , for a specified phase-space region of size ∆. The number, n, of particles is measured inside ∆ and angled brackets . . . denote the average over all events. The factorial moments, along with cumulants [3] and bunching parameters [4], are convenient tools to characterize the multiplicity distributions when ∆ becomes small. Indeed, for uncorrelated particle production within ∆, Poisson statistics holds and F q = 1 for all q. Correlations between particles lead to a broadening of the multiplicity distribution and to dynamical fluctuations. In this case, the normalized factorial moments increase with decreasing ∆. This effect is frequently called intermittency [5].
As a matter of fact, it has been noticed in [5] that the use of factorial moments allows to extract the dynamical signal from the Poisson noise in the analysis of the multiplicity signal in high energy reactions.
In addition, it has been shown that it is possible to define and compute a multi-fractal dimension, D q , for the theory of strong interactions [6,7] F q (∆) = n(n -1) . . . (n -q + 1) / n q ∝ ∆ (q-1)(1-Dq/d)
(1
where d is the dimension of the phase space under consideration (d = 2 for the whole angular phase space, and d = 1 if one has integrated over, say the azimuthal angle). In the constant coupling case D q is well defined and reads
where γ 2 0 = 4C A α S /2π, α S is the strong interaction coupling constant, C A is the gluon color factor [6,7]. The choice of the factorial moments as a specific tool in order to study the scaling behavior of the high energy multiplicity distributions is then useful to analyze the underlying dynamics of the processes. In principle, we can extend this last idea to other fields where factorial moments can be defined.
The multiplicity distribution is defined as P n = σ n / ∞ n=0 σ n , where σ n is the cross section of an n-particle production process (the so-called topological cross section) and the sum is over all possible values of n so that
The generating function can be defined as
which substitutes an analytic function of z in place of the set of numbers P n . Then, we obtain the factorial moment or order q as
and the corresponding definition for cumulants
The expression for G(z) can then be re-written as
ln
The physical meaning of these moments has been discussed in the previous section. Another interpretation can be seen from the above definitions if they are presented in the form of integrals of correlation functions. Let the single symbol y represent all kinematic variables needed to specify the position of each particle in the phase space volume ∆ [8]. A sequence of inclusive q-particle differential cross sections d q σ/dy 1 . . . dy q defines the factorial moments as
Therefore, factorial moments include all correlations within the system of particles under consideration. They represent integral characteristics of any correlation between the particles whereas cumulants of rank q represent genuine q-particle correlations not reducible to the product of lower order correlations.
In Ref. [9], it has been shown that factorial moments can be applied conveniently to quantitative finance. Namely, if we divide price series y(t) in consecutive time windows of lengths ∆, we can define a set of events. In each window, we have a certain number of positive returns n + , where y(t) -y(t -1) > 0, and similarly of negative returns n -. If the sequence of returns is purely uncorrelated, following a Gaussian statistics at all scales, we expect F q = 1 for all q.
As explained in previous sections, correlations returns may lead to a broadening of the multiplicity distributions (n + or n -or even a combination of both) and to dynamical fluctuations. In this case, the factorial moments may increase with decreasing ∆, or increasing the number of bins that divide ∆. In Ref. [9], we have used the factorial moment of second order F 2 for like-sign returns
where n is the average number of positive returns in the full time window (∆), n bins denotes the number of bins in this window and n + k is the number of positive returns in k-th bin. Similarly, we have defined the unlike-sign returns for F 2
with similar notations. Considering different price series, we have shown that for a small resolution in time window [9], below 4 to 8 hours, a deviation with respect to pure Gaussian fluctuations is obse
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