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liminf

Represents the limit inferior (greatest lower bound of limit points) of a sequence or function.

Overview

Essential in advanced mathematical analysis for describing the most conservative limiting behavior of sequences and functions.

  • Particularly important in real analysis and measure theory for examining sequence convergence.
  • Often paired with limsup to establish bounds on limiting behavior.
  • Commonly used in probability theory to analyze the long-term behavior of random processes.
  • Appears frequently in theoretical computer science for analyzing algorithm performance bounds.

Examples

Definition of liminf for a sequence of real numbers.

lim inf⁡n→∞an=sup⁡k≥1inf⁡n≥kan\liminf_{n \to \infty} a_n = \sup_{k \geq 1} \inf_{n \geq k} a_n 
\liminf_{n \to \infty} a_n = \sup_{k \geq 1} \inf_{n \geq k} a_n

Comparing liminf and limsup of a sequence.

lim inf⁡n→∞xn≤lim sup⁡n→∞xn\liminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n 
\liminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n

Using liminf in probability theory for a sequence of events.

P(lim inf⁡n→∞An)≤lim inf⁡n→∞P(An)P(\liminf_{n \to \infty} A_n) \leq \liminf_{n \to \infty} P(A_n) 
P(\liminf_{n \to \infty} A_n) \leq \liminf_{n \to \infty} P(A_n)