TeXipedia

coloncolonsim

Represents an asymptotic equivalence relation, combining a double colon with a tilde to denote a specific mathematical relationship.

Overview

Primarily used in advanced mathematics and theoretical computer science to denote a particular kind of asymptotic relationship between functions or sequences.

  • Common in complexity theory when describing algorithmic behavior
  • Appears in formal proofs and mathematical analysis where precise relationship definitions are needed
  • Often used alongside other asymptotic notation symbols to express detailed comparative relationships
  • Particularly useful in contexts where standard asymptotic notation (like big-O) needs additional refinement

Examples

Defining an asymptotic equivalence relation between functions.

f(x)∷∼g(x) as x→∞f(x) \coloncolonsim g(x) \text{ as } x \to \infty 
f(x) \coloncolonsim g(x) \text{ as } x \to \infty

Expressing probabilistic convergence in distribution.

Xn∷∼X as n→∞X_n \coloncolonsim X \text{ as } n \to \infty 
X_n \coloncolonsim X \text{ as } n \to \infty

Denoting asymptotic similarity in statistical analysis.

θ^n∷∼N(μ,σ2)\hat{\theta}_n \coloncolonsim N(\mu, \sigma^2) 
\hat{\theta}_n \coloncolonsim N(\mu, \sigma^2)