
Contents
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29. Statistical Measures of the Galaxy Distribution 29. Statistical Measures of the Galaxy Distribution
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30. Fair Sample Hypothesis 30. Fair Sample Hypothesis
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31. Two-Point Spatial Correlation Function ξ(r) 31. Two-Point Spatial Correlation Function ξ(r)
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32. Two-Point Correlation Function: Another Definition 32. Two-Point Correlation Function: Another Definition
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33. Two-Point Correlation Function: Poisson Model 33. Two-Point Correlation Function: Poisson Model
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34. Three-Point Correlation Function 34. Three-Point Correlation Function
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35. Four-Point Correlation Function 35. Four-Point Correlation Function
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36. Moments of Counts of Objects 36. Moments of Counts of Objects
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A. Randomly placed cells A. Randomly placed cells
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B. Moments of the counts of neighbors B. Moments of the counts of neighbors
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37. Constraints on ξ and ζ 37. Constraints on ξ and ζ
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38. Probability Generating Function 38. Probability Generating Function
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39. Estimates of PN 39. Estimates of PN
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40. Cluster Model 40. Cluster Model
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41. Power Spectrum 41. Power Spectrum
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42. Power Law Model for the Spectrum 42. Power Law Model for the Spectrum
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43. Bispectrum 43. Bispectrum
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44. Cross Correlation Function 44. Cross Correlation Function
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45. Angular Two-Point Correlation Function 45. Angular Two-Point Correlation Function
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46. Angular Power Spectrum 46. Angular Power Spectrum
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47. Estimating w(θ) 47. Estimating w(θ)
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48. Statistical Uncertainty in the Estimate of w(θ) 48. Statistical Uncertainty in the Estimate of w(θ)
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49. Relation Between Angular and Spatial Two-Point Correlation Functions 49. Relation Between Angular and Spatial Two-Point Correlation Functions
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50. Small Separation Approximation and the Scaling Relation 50. Small Separation Approximation and the Scaling Relation
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51. Decoupling of Magnitude and Position 51. Decoupling of Magnitude and Position
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52. Relation Between ξ and w: Some Examples 52. Relation Between ξ and w: Some Examples
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53. Inversion of the Equation 53. Inversion of the Equation
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54. Angular Three-Point Correlation Function 54. Angular Three-Point Correlation Function
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A. General relation of the reduced spatial and angular functions A. General relation of the reduced spatial and angular functions
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B. Small separation approximation B. Small separation approximation
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C. Model for ζ C. Model for ζ
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D. Methods of estimating z D. Methods of estimating z
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55. Angular Four-Point Correlation Function 55. Angular Four-Point Correlation Function
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56. Correction for Curvature and Expansion 56. Correction for Curvature and Expansion
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57. Summary of Numerical Results 57. Summary of Numerical Results
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58. Power Spectrum of the Extragalactic Light 58. Power Spectrum of the Extragalactic Light
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59. Moments of the Number of Neighbors 59. Moments of the Number of Neighbors
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60. Model for PN 60. Model for PN
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61. Clustering Models 61. Clustering Models
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A. Power law cluster model A. Power law cluster model
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B. Continuous clustering hierarchy B. Continuous clustering hierarchy
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C. Lack of uniqueness C. Lack of uniqueness
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62. Continuous Clustering Hierarchy: Mandelbrot’s Prescription 62. Continuous Clustering Hierarchy: Mandelbrot’s Prescription
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A. Distribution of displacements in the random walk A. Distribution of displacements in the random walk
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B. An unbounded continuous clustering hierarchy B. An unbounded continuous clustering hierarchy
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C. Truncated clustering hierarchy C. Truncated clustering hierarchy
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63. The Mass Correlation Functions 63. The Mass Correlation Functions
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64. Clustering Hierarchy: Continuity Speculation 64. Clustering Hierarchy: Continuity Speculation
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65. Remarks on the Observations 65. Remarks on the Observations
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III n-Point Correlation Functions: Descriptive Statistics
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Published:September 2020
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Abstract
This chapter explores the statistical pattern of the galaxy distribution. It focuses on n-point correlation functions (analogs of the autocorrelation function and higher moments for a continuous function), the descriptive statistics that have proved useful. The approach has also proved useful in many other applications. Of considerable practical importance has been the fact that there is a simple linear equation relating the directly observable angular correlation function to the wanted spatial function. This means the translation from one to the other is fairly easy, and equally important it makes it easy to say how the statistical estimates ought to scale with the depth of the survey and hence to test for possible contamination of the estimates by systematic errors. A third useful result is that the dynamics of the galaxy distribution can be treated in terms of the mass correlation functions: the statistic that proves useful for the reduction of the data may also be useful for the analysis of the theory.
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