Minimum Possible Value of Correlation (Rho)

Introduction

• Problem: Finding the minimum possible value that the correlation (rho) can take for three random variables (x, y, and z) with mutual pairwise correlations.
• Key Concepts: Correlation, Covariance, Positive Semi-Definite Matrices.

Base Concepts and Terminology

• Correlation: Covariance of x and y divided by the product of x and y's standard deviations.
• Covariance: Expectation of the product of centered versions of x and y.
• Formula for Covariance:
  • Expand the product.
  • Apply the linearity of the expectation.
  • Numerator: Product of standard deviations as the square root of the formula for variance.

Key Properties of Correlation

• Correlation is bounded between -1 and 1.
• Correlation Matrix: Matrix with elements as pairwise correlations of variables.
• Positive Semi-Definite Matrix: Correlation matrix is positive semi-definite (proofs available on website).
• Minor and Leading Principle Minor: Determinants of the matrix obtained by deleting rows and columns.
• Property: All leading principle minors of a positive semi-definite matrix are non-negative.

Finding the Minimum Value of Rho

• Task: Find the lower bound and provide an example to show that the bound is attainable.
• Loose bounds: -1 and 1.
• Examples:
  • Independent variables: rho = 0.
  • Pairs with rho = -1: rho = 1 between x and z.
• Interval: Rho is between -1 and 0.
• Using Correlation Matrix Properties:
  • Write the correlation matrix of x, y, and z.
  • Set conditions for the matrix to be positive semi-definite.
  • Compute principal leading minors:
    • First minor is 1, non-negative.
    • Second minor: 1 - rho^2, non-negative for values between -1 and 1.
    • Third minor: Determinant of the matrix itself using Sarrus's rule.
• Key Result: 1 - rho^2 (1 + 2rho) ≥ 0 implies rho >= -1/2.
• Example: Constructing variables with pairwise correlations of -0.5 to show that -0.5 is the minimum value.
  • Example Construction: A1, A2, A3: Independent identically distributed standard uniform random variables.
  • Defining x_i: xi = Ai - Mean(A).
  • Variance of x_i: Calculation shows variance is 2/3.
  • Covariance of x_i and x_k: Calculation shows -1/3.
  • Correlation coefficient: -1/2, showing -0.5 is attainable.

Generalization to n Variables

• Minimum Value of Rho with n Variables: Consider the correlation matrix.
• **Key Results: Det = 1 - rho^(n-1) (1 + (n-1)rho) ≥ 0 implies rho >= -1/(n-1).
• Example Construction for Any n:
  • Using same rationing as before.
  • Variance of x_i: (n-1)/n.
  • Covariance: -1/n -> Correlation: -1/(n-1).
• Consistency: Aligns with results for n=3.
• Conclusion: Minimum correlation converges to zero as n goes to infinity.

Conclusion

• Summary: Found minimum possible value for correlation between three random variables with pairwise correlations and generalized to n variables.
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