Slutsky's theorem

Slutsky's theorem states that if random variables converge in distribution and accompanying variables converge in probability to constants, sums, products, and suitable ratios converge in distribution to the corresponding limits.

Convergence in distribution: A sequence of random variables converges in distribution when its cumulative distribution functions converge at every continuity point of the limit distribution. This is the convergence mode preserved for the transformed random variables.
Random variable: A random variable is a measurable function that assigns numerical values to outcomes of a probability space. The theorem concerns sequences of these objects and their limiting distributions.
Asymptotic statistics: Asymptotic statistics studies the behavior of statistical procedures as sample size grows without bound. Slutsky’s theorem is a standard tool for deriving large-sample distributions.
Eugen Slutsky: Eugen Slutsky was a Ukrainian-Soviet mathematician and economist whose work contributed to probability and mathematical economics. The theorem bears his name and reflects his work on limit behavior.
Convergence in probability: A sequence of random variables converges in probability to a limit when the probability of any fixed positive error tends to zero. The theorem uses this stronger control for variables tending to constants.
Probability measure: A probability measure assigns probabilities to events, with total probability one. Convergence in probability is defined through probabilities assigned to error events.
Wald test: A Wald test evaluates a parameter restriction by scaling an estimator’s deviation by an estimated standard error. Replacing an unknown standard error with a consistent estimate often invokes Slutsky’s theorem.
Central limit theorem: The central limit theorem describes conditions under which normalized sums of random variables converge in distribution to a normal law. Slutsky’s result helps turn central-limit approximations into usable statistical limits.
Continuous mapping theorem: The continuous mapping theorem transfers convergence in distribution through functions continuous at values attained by the limiting variable. It extends the theorem’s limit rules from sums and products to continuous transformations.
Cumulative distribution function: A cumulative distribution function gives the probability that a random variable is no greater than each real number. Its pointwise behavior at continuity points defines convergence in distribution.

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Portmanteau theoremRelated: Portmanteau characterizations underpin distributional limit arguments used in its proof.