Limit Theorem For Statistics Research Articles

Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas

In this paper we study the statistics of combinatorial partitions of the integers, which arise when studying the occupation numbers of loops in the mean field Bose gas. We review the results of Lewis and collaborators and get some more precise estimates on the behavior at the critical point (fluctuations of the condensate component, finite volume corrections to the pressure). We then prove limit shape theorems for the loops occupation numbers. In particular we prove that in a certain range of the parameters, a finite fraction of the total mass is, in the limit, supported by infinitely long loops. We also show that this mass is equal to the mass of the condensed state where all particles have zero momentum.

STATISTICAL PROPERTIES OF ENDOMORPHISMS AND COMPACT GROUP EXTENSIONS

The statistical properties of endomorphisms under the assumption that the associated Perron–Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness. Sufficient conditions are given for quasicompactness of the Perron–Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems for equivariant observations on compact group extensions. Examples considered include compact group extensions of piecewise uniformly expanding maps (for example Lasota–Yorke maps), and subshifts of finite type, as well as systems that are nonuniformly expanding or nonuniformly hyperbolic.

Statistical limit theorems for suspension flows

In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. In this paper, we give conditions under which statistical laws, such as the central limit theorem and almost sure invariance principle, for the underlying discrete time system are inherited by the suspension flow. As a consequence, we give a simpler proof of the results of Ratner (1973) and recover the results of Denker and Philipp (1984) for Axiom A flows. Morcover, we obtain several new results for nonuniformly and partially hyperbolic flows, including frame flows on negatively curved manifolds satisfying a pinching condition.

Power‐law velocity distributions in fracture networks: Numerical evidence and implications for tracer transport

Velocity distributions in two‐ and three‐dimensional networks of discrete fractures are studied through numerical simulations. The distribution of 1/v, where v is the velocity along particle trajectories, is closely approximated by a power law (Pareto) distribution over a wide range of velocities. For the conditions studied, the power law exponents are in the range 1.1–1.8, and generally increase with increasing fracture density. The same is true for the quantity 1/bv, which is related to retention properties of the rock; b is the fracture half‐aperture. Using a stochastic Lagrangian methodology and statistical limit theorems applicable to power‐law variables, it is shown that the distributions of residence times for conservative and reacting tracers are related to one‐sided stable distributions. These results are incompatible with the classical advection dispersion equation and underscore the need for alternative modeling approaches.

Stochastic Prediction of Execution Time for Dynamic Bulk Synchronous Computations

We consider the problem of execution time prediction for non-deterministic multi-phase bulk synchronous computations in multiprocessors. We characterize the computations in two stochastic workload evolution models: additive and multiplicative. The additive model reflects the commutations in which the workload changes between phases are independent of processes' present workload. The multiplicative model becomes relevant when the workload change in a process is proportional to its load base. We take advantage of their salient features and show that conventional approaches based on central limit theorem in statistics are viable to predict the execution time for long run computations. By an elegant coordination of results from order statistics and convergence rates in the central limit theorem, we derive tighter bounds on execution time of short run computations, under some mild assumptions on their workload change distributions. Accuracy of the predictions is analyzed rigorously and verified by simulations.

A noninformetric analysis of the relationship between citation age and journal productivity

A problem, raised by Wallace (JASIS, 37, 136-145, 1986), on the relation between the journal's median citation age and its number of articles is studied. Leaving open the problem as such, we give a statistical explanation of this relationship, when replacing median by mean in Wallace's problem. The cloud of points, found by Wallace, is explained in this sense that the points are scattered over the area in first quadrant, limited by a curve of the form E y= x2 +1, where E is a constant. This curve is obtained by using the Central Limit Theorem in statistics and, hence, has no intrinsic informetric foundation. The article closes with some reflections on explanations of regularities in informetrics, based on statistical, probabilistic or informetric results, or on a combination thereof.

Effect of Noise on Analog Circuit Testing

The effect of test environment noise (tester noise) on test waveforms is considered. We show that tests generated ignoring the tester noise characteristics are prone to failure when actually applied to the circuit-under-test (CUT). The failure may result in the good circuit being declared faulty or the faulty circuit being declared good. This failure is independent of the fault model and nature of the test, i.e., AC or DC, time domain or frequency domain. We characterize the total noise at the primary outputs (PO‘s) of the circuit using second order statistics. We use the noise power spectrum and root mean square (RMS) values to make decisions about the test waveforms and recommend more noise-robust tests. For non-linear circuits we use the Central Limit Theorem of statistics to approximate narrow band noise at a primary input (PI) by a sum of sinusoidal distributions, and we use Monte-Carlo simulations to determine the noise at the PO‘s in the time domain. Results of experiments on an instrumentation amplifier, a biquadratic filter, and a Gilbert multiplier are presented, which prove that valid tests in a noise-free environment are invalid when tester noise is considered.

On the quality of accumulator-based compaction of test responses

The accumulator-based compaction (ABC) technique uses an accumulator to generate a composite fault signature for a circuit under test. The error coverage for this method has been previously analyzed using Markov chains. We describe an alternative technique for calculating the error coverage of ABC using the asymmetric error model. This technique relies on the central limit theorem of statistics and can be applied to other count-based compaction schemes. Our analysis shows that ABC provides very high coverage of asymmetric errors. Experiments on the actual fault coverage for the ISCAS 85 benchmark circuits show that extremely high postcompaction fault coverage (close to 100%) is obtained with ABC. They also indicate that the use of a rotate-carry adder does not always improve the fault coverage; in some cases, the fault coverage is actually reduced.

Theorems as meaningful cultural artifacts: Making the world additive

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the “arithmetic addition” of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions.

Limit Theorems for Statistics that are Partial Sums of Functions of Spacings

Previous article Next article Limit Theorems for Statistics that are Partial Sums of Functions of SpacingsV. P. BorovikovV. P. Borovikovhttps://doi.org/10.1137/1132008PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. Pyke, Spacings. (With discussion.), J. Roy. Statist. Soc. Ser. B, 27 (1965), 395–449 35:7451 0144.41704 Google Scholar[2] D. M. Chibisov, On tests of fit based on sample spacings, Theory Probab. Appl., 3 (1961), 325–329 LinkGoogle Scholar[3] Ronald Pyke, Spacings revisited, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, Univ. California Press, Berkeley, Calif., 1972, 417–427 53:9501 0234.62008 Google Scholar[4] Ulf Grenander, Some direct estimates of the mode, Ann. Math. Statist, 36 (1965), 131–138 30:647 0131.17702 CrossrefGoogle Scholar[5] Lars Holst, Asymptotic normality of sum-functions of spacings, Ann. Probab., 7 (1979), 1066–1072 80j:62021 0421.60017 CrossrefGoogle Scholar[6] Peter Hall, Limit theorems for estimators based on inverses of spacings of order statistics, Ann. Probab., 10 (1982), 992–1003 84a:60034 0516.60024 CrossrefGoogle Scholar[7] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[8] E. V. Khmaladze, Martingale limit theorems for divisible statistics, Theory Probab. Appl., 28 (1983), 530–548 10.1137/1128049 0543.60042 LinkGoogle Scholar[9] R. Sh. Lipster and , A. N. Shiryaev, A functional central limit theorem for semimartingales, Theory Probab. Appl., 25 (1980), 667–688 10.1137/1125084 0471.60038 LinkGoogle Scholar[10] R. M. Mnatsakanov, A martingale limit theorem for additive statistics from sample intervals, Soobshch. Akad. Nauk Gruzin. SSR, 116 (1984), 253–256, (In Russian.) 87c:60034 0562.60024 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the almost sure coverage property of Voronoi tessellation: the ℝ 1 case1 July 2016 | Advances in Applied Probability, Vol. 33, No. 4 Cross Ref Estimation of Functionals of a DensityV. P. Borovikov17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 3AbstractPDF (543 KB) Volume 32, Issue 1| 1988Theory of Probability & Its Applications History Submitted:10 June 1985Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132008Article page range:pp. 86-97ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Martingale Limit Theorems for Divisible Statistics

Previous article Next article Martingale Limit Theorems for Divisible StatisticsE. V. KhmaladzeE. V. Khmaladzehttps://doi.org/10.1137/1128049PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. I. Medvedev, Some theorems on the asymptotic distribution of the $\chi \sp{2}$ statistic, Dokl. Akad. Nauk SSSR, 192 (1970), 987–989, (In Russian.) 42:1263 Google Scholar[2] V. F. Kolchin, , B. A. Sevast'yanov and , V. P. Chistyakov, Random allocations, V. H. Winston & Sons, Washington, D.C., 1978xi+262 57:10758b Google Scholar[3] Yu. K. Orlov and , R. Ya. Chitashlivi, On the statistical significance of the Zipf laws, Soobshcheniya AN GrSSR, 108 (1982), 513–516, (In Russian.) Google Scholar[4] Yu. I. Medvedev, Separable statistics in a polynomial scheme, I, II, Theory Prob. Appl., 22 (1977), 1–15, 22 (1977), pp. 607–615 LinkGoogle Scholar[5] G. I. Ivchenko and , Yu. I. Medvedev, Decomposable statistics and hypothesis testing for grouped data, Theory Prob. Appl., 25 (1980), 540–551 0494.62052 LinkGoogle Scholar[6] C. Morris, Central limit theorems for multinomial sums, Ann. Statist., 3 (1975), 165–188 51:7094 0305.62013 CrossrefGoogle Scholar[7] R. Sh. Liptser and , A. N. Shiryaev, The functional central limit theorem f or semi-martingaless, Theory Prob. Appl., 25 (1980), 680–703 Google Scholar[8] P. Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[9] R. Sh. Liptser and , A. N. Shiryaev, Statistics of Random Processes, Springer, New York, 1977–78 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails The probabilities of large deviations for a certain class of statistics associated with multinomial distribution9 October 2020 | ESAIM: Probability and Statistics, Vol. 24 Cross Ref A Conversation with Estate V. KhmaladzeStatistical Science, Vol. 31, No. 3 Cross Ref Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law14 July 2016 | Journal of Applied Probability, Vol. 48, No. 4 Cross Ref Martingale limit theorems of divisible statistics in a multinomial scheme with mixed frequenciesStatistics & Probability Letters, Vol. 81, No. 8 Cross Ref Estimation of Functionals of a DensityV. P. Borovikov17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 3AbstractPDF (543 KB)Tests of Homogeneity under Small Grouping. Martingale Limit TheoremsI. K. Urinov17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 4AbstractPDF (1268 KB)Goodness-of-Fit Tests for Generalized Urn Schemes Based on Separable StatisticsA. F. Ronzhin17 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 1AbstractPDF (980 KB)Limit Theorems for Statistics that are Partial Sums of Functions of SpacingsV. P. Borovikov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 1AbstractPDF (904 KB)On the Convergence of Separable Statistics to a Wiener ProcessR. M. Mnatsakanov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 1AbstractPDF (438 KB)On the Convergence of Divisible Statistics to the Poisson ProcessR. M. Mnatsakanov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 2AbstractPDF (537 KB)On a Method for Constructing Goodness-of-Fit Tests for Parametric Hypotheses in $R^m$A. M. Nikabadze17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 3AbstractPDF (498 KB)A Functional Limit Theorem for Additively Separable Statistics in the Case of Very Rare EventsR. M. Mnatsakanov17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 3AbstractPDF (424 KB)Estimates of Closeness of the Distribution of a Randomized Separable Statistic to the Normal Law in a Multinomial SchemeSh. A. Mirakhmedov28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (472 KB) Volume 28, Issue 3| 1984Theory of Probability & Its Applications History Submitted:14 April 1982Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128049Article page range:pp. 530-548ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On U-statistics and v. mise? statistics for weakly dependent processes

Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v. Mises' functional are established when the underlying processes are not necessarily independent. We consider absolutely regular processes [24] and processes (X n)n≧1 which are uniformly mixing [14] as well as their time reversal (X −n )n≦−1, called uniformly mixing in both directions of time. Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and a.s. results in the absolutely regular and uniformly mixing case. Here we extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorems and a.s. approximation by a Brownian motion. Extensions to multisample versions and other extensions are briefly discussed.

Crushing of single irregularly shaped particles by compression: size distribution of progeny particles

On the basis of experimental results with quartzite an analytical description is presented for the progeny size distribution of irregularly shaped particles crushed singly by slow compression. For the four initial size ranges investigated the progeny size distributions are composed of four truncated (three-parameter) log-normal distributions for each comminution energy range. The dependence of the parameters of the experimental size distribution upon the comminution energy has been determined. In contrast to the results for single crushed spheres, for irregularly shaped initial particles the ratio of newly formed surface area to comminution energy depends upon the comminution energy. This is caused by the influence of friction during the crushing of the initial particles. On the basis of the central limit theorem of statistics an attempt is made to explain the origin of the mathematical form of the progeny size distribution produced by the crushing of spheres. The genesis of the progeny size distribution is also qualitatively established for irregularly shaped initial particles. On the same theoretical basis one may reasonably expect that the size distribution of progeny particles is truncated log-normal. In the case of the progeny of irregular parent particles, as distinct from spheres, several such distributions should be superimposed on one another.

Some exact and approximate solutions of the nonlinear Boltzmann equation with applications to aerosol coagulation

The nonlinear kinetic equation describing aerosol coagulation and deposition is studied analytically by a new method based upon successive coalescences. The method is illustrated by using a constant coagulation kernel and a deposition term which depends upon a power of the particle volume (zero or unity). The success of the method rests upon a novel use of the central limit theorem of statistics applied to causal systems. The techniques are of general interest in the solution of the nonlinear Boltzmann equation in the kinetic theory of gases and in the case studied show that the volume distribution of successive generations of coalescences tends rapidly to a gamma distribution. Closed form solutions are obtained with a constant coagulation kernel and a deposition rate that is proportional to any power of particle volume. Numerical results are obtained for several cases which indicate that the values m=0 and 1 for the deposition rate are not unreasonable for the purposes of general comment.

A limit theorem for statistics of spatial data

A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate ‘ergodic’ distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the ‘true’ one. Examples in mathematical morphology are given.

A limit theorem for statistics of spatial data

A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate ‘ergodic’ distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the ‘true’ one. Examples in mathematical morphology are given.

Some Limit Theorems in Statistics.

Some Limit Theorems in Statistics.

Book Reviews

Abstract Robust Estimates of Location: Survey and Advances, D.F. Andrews, P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers, and J.W. Tukey. Princeton: Princeton University Press, 1972. ix+373 pp. $12.50 cloth; $6.00 paperback. Reviewed by David C. Hoaglin Nineteenth-Century Society, E.A. Wrigley (ed.). New York: Cambridge University Press, 1972. vii+448 pp. $27.50. Reviewed by Donald N. McCloskey The Wages of War 1816–1965: A Statistical Handbook, J. David Singer and Melvin Small. New York: John Wiley and Sons, Inc., 1972. xii+419 pp. $13.95. Reviewed by Donald N. McCloskey The Working of Econometric Models, M. Morishima, Y. Murata, T. Nouse, and M. Saito. New York: Cambridge University Press, 1972. ix+339 pp. $22.50. Reviewed by Michael D. McCarthy On the Measurement of Human Fertility: Selected Writings of Louis Henry, Mindel C. Sheps and Evelyne Lapierre-Adamcyk (eds.). New York: American Elsevier Publishing Co., Inc., 1972. xix+228 pp. $15.00. Reviewed by Zenas M. Sykes Introduction to Bayesian Inference and Decision, Robert L. Winkler. New York: Holt, Rinehart, and Winston, Inc., 1972. xi+563 pp. $12.50. Reviewed by David K. Hildebrand Uncertainty, Calibration and Probability, C.F. Dietrich New York: John Wiley and Sons, Inc., 1973. xii-411 pp. $49.00. Reviewed by Robert Hooke Introduction to Linear Models, Joe H. Ward, Jr. and Earl Jennings. Englewood Cliffs: Prentice-Hall, Inc., 1973. xviii+333 pp. $10.95. Reviewed by John J. Bartko Contiguity of Probability Measures: Some Applications in Statistics, George G. Roussas. New York: Cambridge University Press, 1972. xiii+248 pp. $21.00. Reviewed by J. Wolfowitz Comment Reviewed by Jaroslav Hájek Reply Reviewed by J. Wolfowitz Statistical Analysis: A Computer Oriented Approach, A.A. Afifi and S.P. Azen. New York: Academic Press, Inc., 1972. xviii+366 pp. $12.50. Reviewed by David F. Andrews Statistical Inference under Order Restrictions, R.E. Barlow, D.J. Bartholomew, J.M. Bremner and H.D. Brunk. New York: John Wiley and Sons, Inc., 1972. xii+388 pp. $21.95. Reviewed by Edward J. Wegman Some Limit Theorems in Statistics, R.R. Bahadur. Philadelphia: Society for Industrial and Applied Mathematics, 1971. v+42 pp. $4.20 paperback. Reviewed by Prem K. Goel Weak Convergence of Measures: Applications in Probability, Patrick Billingsley. Philadelphia: Society for Industrial and Applied Mathematics, 1971. v+31 pp. $2.80 paperback. Reviewed by Prem K. Goel Statistics, Roger Carlson. San Francisco: Holden-Day, Inc., 1973. xiv+393 pp. $11.95. Reviewed by James H. Ware Bibliography on Experiment and Treatment Design, Pre-1968, Leslie N. Balaam and Walter T. Federer. New York: Hafner Publishing Company, Inc., 1973. 757. pp. $27.50. Reviewed by Max Halperin Probability and Statistical Inference for Scientists and Engineers, Isaac N. Gibra. Englewood Cliffs: Prentice-Hall Inc., 1973. xii+596 pp. $16.50. Reviewed by Roger H. Moore Staff Reviews: Problems and Issues in Current Econometric Practice, Karl Brunner (ed.). Columbus, Ohio: College of Administrative Science, Ohio State University, 1973. vii+284 pp. $10.00 Staff Reviews: Elementary Statistics, Robert Johnson. North Scituate, Massachusetts: Duxbury Press, 1973. xvi+480 pp. $10.00 Staff Reviews: Fundamentals of Systems Engineering: Probabilistic Models, Tung Au, Richard M. Shane, and Lester A. Hoel. Reading, Massachusetts: Addison-Wesley Publishing Co., 1972. vii+375 pp. $14.95 Staff Reviews: Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta (Eds.). New York: Springer-Verlag, 1972. viii+492 pp. $8.60 paperback Staff Reviews: Branching Processes, K.B. Athreya and P.E. Ney. New York: Springer-Verlag New York Inc., 1972. xi+287 pp. $24.10 Staff Reviews: Inference and Decision, D.A.S. Fraser (ed.). New York: John Wiley and Sons, Inc., 1973. vi+82 pp. $13.75 Staff Reviews: The Statistics Cum Index, James L. Dolby and John W. Tukey. Los Altos, California: The R and D Press, 1973. xviii+498 pp. $28.80 Staff Reviews: Index to Statistics and Probability: Citation Index, John W. Tukey. Los Altos, California: The R and D Press, 1973. xxx+1269 pp. $60.00 Staff Reviews: Collected Papers of R. A. Fisher: Volume I, 1912–24, and Volume II, 1925–31, J.H. Bennett (ed.). Adelaide, South Australia: The University of Adelaide, 1971 and 1972. 604 pp and 558 pp. $20.00 per volume

A calculation of upper bounds on time-decay constants in pulsed neutron experiments

Numerical values of (νΣ)min have been obtained for a liquid and a crystal in the incoherent approximation. The liquid calculations are based on the effective width model and results for water are given. The crystal results are obtained from the Gaussian approximation, with multi-phonon terms approximated by the central limit theorem of statistics. Beryllium and graphite have been studied using the Debye and Egelstaff phonon frequency spectra, respectively. The limitations of the heavy crystal model are stressed.

On a classical statistical model with the nonrelativistic integrals of motion

The classical nonrelativistic phase-space integral for fixed energy, momentum, angular momentum and center of mass is evaluated for large particle numbers by means of the central limit theorem of statistics. The problem is treated covariantly with respect to all transformations of the Galilei group. As result we get Ωs as function of the invariants corresponding to the c.m.s. energy E0 and angular momentumL 0 in the formΩ s (E 0,L 0 2 )=Ω s (E 0)F(L 0 2 ,E,0).Ω(E 0) is the well-known phase-space at fixed energy, momentum and center of mass, andF(E 0,L 0 2 =[3/4πmR 2 E 0]3/2 exp[-(3L 0 2 /4mR 2 E 0)] is a normalized probability density for the angular momentum L 0 2 .