Slutsky’s Theorem and Continuous Mapping

Combining limiting random quantities with consistent plug-in estimates

Statistics

Asymptotics

Inference

Continuous mapping and Slutsky’s theorem, with conditions, proof intuition, studentization, and failure cases.

Author

Jiahong Sun

Published

September 19, 2026

Modified

September 19, 2026

1 Two Rules for Building Useful Limits

Asymptotic arguments rarely end with the first limiting distribution. Analysts transform estimators, replace unknown nuisance parameters by consistent estimates, and combine several random quantities. The continuous mapping theorem handles continuous transformations; Slutsky’s theorem makes a particularly important combination valid when one component converges in probability to a constant.

These are limit theorems, not declarations that estimation error is absent. They say when the extra random error is asymptotically negligible on the scale being studied.

2 Continuous Mapping Theorem

Let $X_n\xrightarrow{d}X$, and let $g$ be measurable. If $g$ is continuous at $X$ with probability one—equivalently, the limit assigns probability zero to the discontinuity set $D_g$—then

\[ \boxed{g(X_n)\xrightarrow{d}g(X).} \]

The theorem also has probability and almost-sure forms: if $X_n\xrightarrow{P}X$ or $X_n\xrightarrow{a.s.}X$, the corresponding convergence is preserved under continuity at the relevant limiting values.

Continuity everywhere is sufficient but stronger than necessary. For the threshold map $g(x)=\mathbf 1(x\leq a)$, the only discontinuity is $a$. Thus $X_n\xrightarrow{d}X$ implies $P(X_n\leq a)\to P(X\leq a)$ only when $P(X=a)=0$, matching the CDF definition of convergence in distribution.

3 Slutsky’s Theorem

Suppose

\[ X_n\xrightarrow{d}X, \qquad Y_n\xrightarrow{P}c, \]

where $c$ is a constant. Then $(X_n,Y_n)\xrightarrow{d}(X,c)$. Applying continuous functions to this joint limit gives

\[ \boxed{ \begin{aligned} X_n+Y_n &\xrightarrow{d} X+c,\\ X_nY_n &\xrightarrow{d} cX,\\ X_n/Y_n &\xrightarrow{d} X/c &&\text{if }c\neq0. \end{aligned}} \]

No independence between $X_n$ and $Y_n$ is required. The constant limit is the crucial feature: although $Y_n$ remains random at finite $n$, it collapses to a fixed value asymptotically.

A useful proof intuition is that $Y_n=c+o_p(1)$. Joint convergence with $X_n$ follows because the second coordinate’s displacement vanishes in probability; addition, multiplication, and division then follow from continuous mapping. Division is continuous only away from a zero denominator.

4 Worked Example: Studentization

Let $X_1,\ldots,X_n$ be iid with mean $\mu$ and finite, positive variance $\sigma^2$. The central limit theorem gives

\[ \frac{\sqrt n(\bar X_n-\mu)}{\sigma} \xrightarrow{d}N(0,1). \]

Because the sample standard deviation $S_n\xrightarrow{P}\sigma$, its ratio satisfies $S_n/\sigma\xrightarrow{P}1$. Slutsky’s theorem therefore yields

\[ \boxed{ \frac{\sqrt n(\bar X_n-\mu)}{S_n} = \frac{\sqrt n(\bar X_n-\mu)/\sigma}{S_n/\sigma} \xrightarrow{d}N(0,1). } \]

This justifies replacing the unknown $\sigma$ by a consistent estimate in the asymptotic pivot. It does not say the statistic has an exact normal distribution. Under normal sampling it has an exact Student’s $t_{n-1}$ distribution; under broader finite-variance sampling, normality is only a large-sample approximation.

If the $X_i$ are summaries from independently sampled biological specimens, $n$ counts specimens. Multiple reads, cells, or technical runs within a specimen do not become independent $X_i$ values merely because there are many of them. A correct algebraic limit cannot rescue a misspecified unit of replication.

5 What the Theorems Do Not Permit

5.1 Separate marginal limits are not a joint limit

If $X_n\xrightarrow{d}X$ and $Y_n\xrightarrow{d}Y$, the limit of $X_n+Y_n$ is not determined without information about their joint behavior. For example, two standard-normal marginals can be identical, independent, or perfectly negatively related; their sum variances are then respectively $4$, $2$, or $0$ under suitable couplings. Slutsky avoids this ambiguity because one limiting component is constant.

5.2 Discontinuities at mass points matter

Let $X_n=1/n$ and $X=0$. Then $X_n\to X$, but for $g(x)=\mathbf 1(x>0)$, $g(X_n)=1$ while $g(X)=0$. The map is discontinuous exactly where the limiting variable lies. “Almost surely continuous under the limit” is a real condition, not a technical decoration.

5.3 A denominator converging to zero is unsafe

From $Y_n\xrightarrow{P}0$, one cannot conclude a stable ratio limit for $X_n/Y_n$. The ratio may diverge or depend on relative rates. Ratios of weak signals and estimators near parameter boundaries require a dedicated joint analysis.

6 Conditions, Diagnostics, and Finite-Sample Limits

Claim	Required condition	Failure or check
Transform a weak limit	$g$ is continuous with probability one under the limiting law.	Locate discontinuities and check whether the limit has atoms there.
Add or multiply a plug-in quantity	The plug-in converges in probability to a constant.	Consistency can fail under misspecification, dependence, or weak identification.
Divide by a plug-in quantity	Its constant limit is nonzero.	Inspect whether the denominator can approach or cross zero.
Combine multiple nondegenerate limits	Establish joint convergence, not just marginal convergence.	Preserve dependence and covariance in the limiting vector.
Use the limit at the observed $n$	The approximation must be adequate for the needed region, especially the tails.	Compare with an exact law, design-faithful simulation, or bootstrap when justified.

These theorems provide no universal sample-size threshold or convergence rate. A nuisance estimator can be consistent yet too noisy at the current sample size for a plug-in approximation to work well. Tail probabilities, ratios, and near-discontinuous decisions are especially sensitive.

7 Role in Asymptotic Statistics

The delta method combines a Taylor expansion with Slutsky’s theorem to discard a smaller-order remainder. Consistent variance estimates turn asymptotically normal estimation errors into studentized statistics. Together, these tools form the bridge from a theoretical limit involving unknown parameters to an implementable large-sample approximation.

The scientific interpretation remains tied to the estimand and sampling design. Slutsky’s theorem licenses a statistical substitution; it does not show that a measured association is causal, practically important, or biologically reproducible.

8 Connections and Take-home Messages

Convergence in probability and almost-sure convergence defines the modes used by consistent plug-in estimates.
Convergence in distribution gives the weak limit transformed by these results.
Central limit theorem supplies many initial normal limits.
Delta method is a central application.

Continuous mapping preserves convergence through functions that are continuous where the limiting variable lives. Slutsky permits consistent nuisance estimates to replace constants in sums, products, and nonzero-denominator ratios. Neither theorem supplies joint convergence, finite-sample accuracy, or a valid experimental design automatically.

9 References

van der Vaart, A. W. Asymptotic Statistics. Cambridge University Press, 1998, Chapters 2–3.
Billingsley, P. Convergence of Probability Measures, 2nd ed. Wiley, 1999.
MIT OpenCourseWare. Statistics for Applications, Lecture 1: Introduction, including Slutsky’s theorem. Lecture slides.
MIT OpenCourseWare. Mathematical Statistics, Lecture 16: Asymptotics—Consistency and Delta Method. Lecture notes.

--- title: "Slutsky's Theorem and Continuous Mapping" subtitle: "Combining limiting random quantities with consistent plug-in estimates" description: "Continuous mapping and Slutsky's theorem, with conditions, proof intuition, studentization, and failure cases." author: "Jiahong Sun" date: "2026-09-19" date-modified: "2026-09-19" categories: [Statistics, Asymptotics, Inference] toc: true number-sections: true draft: false --- # Two Rules for Building Useful Limits Asymptotic arguments rarely end with the first limiting distribution. Analysts transform estimators, replace unknown nuisance parameters by consistent estimates, and combine several random quantities. The **continuous mapping theorem** handles continuous transformations; **Slutsky's theorem** makes a particularly important combination valid when one component converges in probability to a constant. These are limit theorems, not declarations that estimation error is absent. They say when the extra random error is asymptotically negligible on the scale being studied. # Continuous Mapping Theorem Let $X_n\xrightarrow{d}X$, and let $g$ be measurable. If $g$ is continuous at $X$ with probability one---equivalently, the limit assigns probability zero to the discontinuity set $D_g$---then $$ \boxed{g(X_n)\xrightarrow{d}g(X).} $$ The theorem also has probability and almost-sure forms: if $X_n\xrightarrow{P}X$ or $X_n\xrightarrow{a.s.}X$, the corresponding convergence is preserved under continuity at the relevant limiting values. Continuity everywhere is sufficient but stronger than necessary. For the threshold map $g(x)=\mathbf 1(x\leq a)$, the only discontinuity is $a$. Thus $X_n\xrightarrow{d}X$ implies $P(X_n\leq a)\to P(X\leq a)$ only when $P(X=a)=0$, matching the CDF definition of [convergence in distribution](convergence-in-distribution.qmd). # Slutsky's Theorem Suppose $$ X_n\xrightarrow{d}X, \qquad Y_n\xrightarrow{P}c, $$ where $c$ is a constant. Then $(X_n,Y_n)\xrightarrow{d}(X,c)$. Applying continuous functions to this joint limit gives $$ \boxed{ \begin{aligned} X_n+Y_n &\xrightarrow{d} X+c,\\ X_nY_n &\xrightarrow{d} cX,\\ X_n/Y_n &\xrightarrow{d} X/c &&\text{if }c\neq0. \end{aligned}} $$ No independence between $X_n$ and $Y_n$ is required. The constant limit is the crucial feature: although $Y_n$ remains random at finite $n$, it collapses to a fixed value asymptotically. A useful proof intuition is that $Y_n=c+o_p(1)$. Joint convergence with $X_n$ follows because the second coordinate's displacement vanishes in probability; addition, multiplication, and division then follow from continuous mapping. Division is continuous only away from a zero denominator. # Worked Example: Studentization Let $X_1,\ldots,X_n$ be iid with mean $\mu$ and finite, positive variance $\sigma^2$. The [central limit theorem](central-limit-theorem.qmd) gives $$ \frac{\sqrt n(\bar X_n-\mu)}{\sigma} \xrightarrow{d}N(0,1). $$ Because the sample standard deviation $S_n\xrightarrow{P}\sigma$, its ratio satisfies $S_n/\sigma\xrightarrow{P}1$. Slutsky's theorem therefore yields $$ \boxed{ \frac{\sqrt n(\bar X_n-\mu)}{S_n} = \frac{\sqrt n(\bar X_n-\mu)/\sigma}{S_n/\sigma} \xrightarrow{d}N(0,1). } $$ This justifies replacing the unknown $\sigma$ by a consistent estimate in the asymptotic pivot. It does not say the statistic has an exact normal distribution. Under normal sampling it has an exact Student's $t_{n-1}$ distribution; under broader finite-variance sampling, normality is only a large-sample approximation. If the $X_i$ are summaries from independently sampled biological specimens, $n$ counts specimens. Multiple reads, cells, or technical runs within a specimen do not become independent $X_i$ values merely because there are many of them. A correct algebraic limit cannot rescue a misspecified unit of replication. # What the Theorems Do Not Permit ## Separate marginal limits are not a joint limit If $X_n\xrightarrow{d}X$ and $Y_n\xrightarrow{d}Y$, the limit of $X_n+Y_n$ is not determined without information about their joint behavior. For example, two standard-normal marginals can be identical, independent, or perfectly negatively related; their sum variances are then respectively $4$, $2$, or $0$ under suitable couplings. Slutsky avoids this ambiguity because one limiting component is constant. ## Discontinuities at mass points matter Let $X_n=1/n$ and $X=0$. Then $X_n\to X$, but for $g(x)=\mathbf 1(x>0)$, $g(X_n)=1$ while $g(X)=0$. The map is discontinuous exactly where the limiting variable lies. “Almost surely continuous under the limit” is a real condition, not a technical decoration. ## A denominator converging to zero is unsafe From $Y_n\xrightarrow{P}0$, one cannot conclude a stable ratio limit for $X_n/Y_n$. The ratio may diverge or depend on relative rates. Ratios of weak signals and estimators near parameter boundaries require a dedicated joint analysis. # Conditions, Diagnostics, and Finite-Sample Limits | Claim | Required condition | Failure or check | |---|---|---| | Transform a weak limit | $g$ is continuous with probability one under the limiting law. | Locate discontinuities and check whether the limit has atoms there. | | Add or multiply a plug-in quantity | The plug-in converges in probability to a constant. | Consistency can fail under misspecification, dependence, or weak identification. | | Divide by a plug-in quantity | Its constant limit is nonzero. | Inspect whether the denominator can approach or cross zero. | | Combine multiple nondegenerate limits | Establish joint convergence, not just marginal convergence. | Preserve dependence and covariance in the limiting vector. | | Use the limit at the observed $n$ | The approximation must be adequate for the needed region, especially the tails. | Compare with an exact law, design-faithful simulation, or bootstrap when justified. | These theorems provide no universal sample-size threshold or convergence rate. A nuisance estimator can be consistent yet too noisy at the current sample size for a plug-in approximation to work well. Tail probabilities, ratios, and near-discontinuous decisions are especially sensitive. # Role in Asymptotic Statistics The [delta method](delta-method.qmd) combines a Taylor expansion with Slutsky's theorem to discard a smaller-order remainder. Consistent variance estimates turn asymptotically normal estimation errors into studentized statistics. Together, these tools form the bridge from a theoretical limit involving unknown parameters to an implementable large-sample approximation. The scientific interpretation remains tied to the estimand and sampling design. Slutsky's theorem licenses a statistical substitution; it does not show that a measured association is causal, practically important, or biologically reproducible. # Connections and Take-home Messages - [Convergence in probability and almost-sure convergence](convergence-in-probability-and-almost-sure-convergence.qmd) defines the modes used by consistent plug-in estimates. - [Convergence in distribution](convergence-in-distribution.qmd) gives the weak limit transformed by these results. - [Central limit theorem](central-limit-theorem.qmd) supplies many initial normal limits. - [Delta method](delta-method.qmd) is a central application. Continuous mapping preserves convergence through functions that are continuous where the limiting variable lives. Slutsky permits consistent nuisance estimates to replace constants in sums, products, and nonzero-denominator ratios. Neither theorem supplies joint convergence, finite-sample accuracy, or a valid experimental design automatically. # References - van der Vaart, A. W. *Asymptotic Statistics*. Cambridge University Press, 1998, Chapters 2--3. - Billingsley, P. *Convergence of Probability Measures*, 2nd ed. Wiley, 1999. - MIT OpenCourseWare. *Statistics for Applications*, Lecture 1: Introduction, including Slutsky's theorem. [Lecture slides](https://ocw.mit.edu/courses/18-650-statistics-for-applications-fall-2016/7836ea6d101679d495194bc3c5f227fa_MIT18_650F16_Introduction.pdf). - MIT OpenCourseWare. *Mathematical Statistics*, Lecture 16: Asymptotics---Consistency and Delta Method. [Lecture notes](https://ocw.mit.edu/courses/18-655-mathematical-statistics-spring-2016/resources/mit18_655s16_lecnote16/).