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plim

Denotes the probability limit operator in statistics and probability theory, indicating the convergence in probability of a sequence of random variables.

Overview

Essential in advanced statistical analysis and econometrics for describing the asymptotic behavior of estimators and random sequences.

  • Commonly used in proving consistency of statistical estimators
  • Appears frequently in time series analysis and large sample theory
  • Distinguished from regular limits by considering probabilistic convergence
  • Important in theoretical foundations of maximum likelihood estimation and regression analysis

Examples

Probability limit of a sequence of random variables.

plim⁡n→∞Xˉn=μ\plim_{n \to \infty} \bar{X}_n = \mu 
\plim_{n \to \infty} \bar{X}_n = \mu

Probability limit in an econometric estimator consistency proof.

plim⁡n→∞β^n=β0\plim_{n \to \infty} \hat{\beta}_n = \beta_0 
\plim_{n \to \infty} \hat{\beta}_n = \beta_0

Probability limit showing convergence in probability.

plim⁡T→∞1T∑t=1TXt=E[X]\plim_{T \to \infty} \frac{1}{T}\sum_{t=1}^T X_t = E[X] 
\plim_{T \to \infty} \frac{1}{T}\sum_{t=1}^T X_t = E[X]