Understanding the Maximum Covariance Value for Two Arbitrary Sets

Maximum Covariance Value

Covariance is a measure of how two variables change together. Given two random variables (X) and (Y), the covariance is defined as:

(text{Cov}(X, Y) E[(X - mu_X)(Y - mu_Y)]),

where (mu_X) and (mu_Y) are the means of (X) and (Y), respectively.

Properties of Covariance and Variance

There are several important properties of covariance and variance worth noting:

(text{Cov}(aX, Y) a cdot text{Cov}(X, Y)), (text{Var}(X) text{Cov}(X, X) geq 0), (text{Cov}(X, Y) text{Cov}(Y, X)), (text{Cov}(aX, bY) ab cdot text{Cov}(X, Y)).

Using these properties, we can derive the maximum value of the covariance between two sets (X) and (Y).

Deriving the Bound on Covariance

Consider the sets (X) and (Y). Define a new variable (Z):

(Z X - frac{text{Cov}(X, Y)}{text{Var}(Y)}Y)

The variance of (Z) can be calculated as follows:

(text{Var}(Z) text{Cov}(Z, Z) text{Cov}left(X - frac{text{Cov}(X, Y)}{text{Var}(Y)}Y, X - frac{text{Cov}(X, Y)}{text{Var}(Y)}Yright))

(text{Var}(Z) text{Cov}(X, X) - 2 text{Cov}left(X, frac{text{Cov}(X, Y)}{text{Var}(Y)}Y right) left(frac{text{Cov}(X, Y)}{text{Var}(Y)}right) text{Cov}(Y, Y))

(text{Var}(Z) text{Var}(X) - 2 frac{text{Cov}(X, Y)^2}{text{Var}(Y)} text{Var}(Y))

Since (text{Var}(Z) geq 0), we can set the expression to be non-negative:

(text{Var}(X) - frac{text{Cov}(X, Y)^2}{text{Var}(Y)} text{Var}(Y) geq 0)

Simplifying the above inequality:

(text{Cov}(X, Y)^2 leq text{Var}(X) cdot text{Var}(Y))

Taking the square root of both sides, we get:

(text{Cov}(X, Y) leq sqrt{text{Var}(X) cdot text{Var}(Y)})

Conclusion

The maximum value of the covariance between two sets (X) and (Y), given the properties of covariance and variance, is bounded by the product of the square roots of their variances:

(text{Cov}(X, Y) leq sqrt{text{Var}(X) cdot text{Var}(Y)})

This result indicates that the maximum value of the covariance is directly related to the variances of the two sets. This relationship is crucial in understanding and interpreting the co-movements between two sets of data.