Slutsky's & Continuous Mapping Theorems — Slutsky's theorem and continuous mapping theorem (intro) - Statistics Graduate Level

SECTION 1

Historical Context & Motivation

The study of convergence of random variables matured rapidly in the early twentieth century as probability theory transitioned from an informal collection of heuristics into a rigorous mathematical discipline. A persistent challenge was understanding how operations on sequences of random variables—sums, products, and general transformations—interact with different modes of convergence. Two results that answered this question in elegant, complementary ways are Slutsky's theorem and the continuous mapping theorem. Together they form the algebraic backbone of asymptotic statistics, allowing practitioners to derive the limiting distributions of complex estimators from simpler building blocks.

Before these theorems were formalized, statisticians relied on ad hoc arguments to justify asymptotic approximations. For instance, showing that a sample mean converges in distribution is only the beginning: in practice we often divide by a sample standard deviation, add a bias correction, or apply a nonlinear link function. Without a systematic rule explaining how convergence in distribution behaves under arithmetic operations and continuous transformations, each new estimator required a bespoke proof—a situation that was both tedious and error-prone.

1905 • Foundations of Measure-Theoretic Probability

Henri Lebesgue's integration theory provided the measure-theoretic framework that would underpin rigorous definitions of convergence in distribution and in probability.

1925 • Slutsky's Seminal Paper

Eugen (Evgeny) Slutsky published results showing how combining sequences converging in probability with sequences converging in distribution preserves the latter mode of convergence, laying the groundwork for what we now call Slutsky's theorem.

1933 • Kolmogorov's Axioms

Andrey Kolmogorov published his axiomatization of probability, creating the formal setting in which convergence modes (a.s., in probability, in distribution) could be precisely distinguished.

1947–1956 • Continuous Mapping Theorem Formalized

Samuel S. Wilks (1947), Alexander Skorokhod, and later Patrick Billingsley codified the continuous mapping theorem, showing that any continuous function preserves convergence in distribution, in probability, and almost surely.

1968 • Billingsley's Convergence of Probability Measures

Patrick Billingsley's influential monograph unified the treatment of weak convergence and provided the modern presentation of both theorems used in graduate courses today.

The central question these theorems address can be stated simply: if we know that a random variable X_n converges (in some sense) to X, and we apply an operation—adding a term, multiplying, or evaluating a function g—what can we say about the convergence of the resulting sequence? Slutsky's theorem handles the case where one operand is converging to a constant, while the continuous mapping theorem handles the case of applying a continuous function.

SECTION 2

Core Principles & Definitions

Before stating the two theorems, we need to recall three modes of convergence that form the hierarchy at the heart of asymptotic theory. Understanding the relationships among these modes is essential because each theorem's hypotheses specify precisely which mode of convergence is required for each input sequence.

Convergence in Distribution (⇒d)

X_n →^d X means F_n(x) → F(x) at every continuity point x of F. This is the weakest of the three modes and is the typical conclusion of the Central Limit Theorem.

Convergence in Probability (⇒p)

X_n →^p X means P(|X_n − X| > ε) → 0 for every ε > 0. This is stronger than convergence in distribution and is the typical conclusion of the Weak Law of Large Numbers.

Almost Sure Convergence (⇒a.s.)

X_n →^a.s. X means P(ω : X_n(ω) → X(ω)) = 1. The strongest mode; implies convergence in probability, which in turn implies convergence in distribution.

Slutsky's Theorem — Key Idea

If X_n →^d X and Y_n →^p c (a constant), then X_n + Y_n →^d X + c, and similarly for products and ratios.

Continuous Mapping Theorem — Key Idea

If X_n → X in any of the three modes and g is a continuous function, then g(X_n) → g(X) in the same mode. Continuity may fail on a set of probability zero under the limit measure.

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KEY TAKEAWAY

Think of convergence in distribution as a fragile package: you cannot just add or multiply two such sequences and expect the result to converge nicely. Slutsky's theorem says the package is safe to combine with something that is essentially becoming a known constant—like taping a rigid label onto the box. The continuous mapping theorem says the package survives any smooth handling (continuous function). Rough handling (discontinuous functions) can break the convergence, just as dropping the package might break the contents.

SECTION 3

Visual Explanation — Convergence Hierarchy & Theorem Flow

The diagram below illustrates the logical relationship between the three modes of convergence and shows exactly where Slutsky's theorem and the continuous mapping theorem fit into the picture. Notice that the continuous mapping theorem preserves whichever mode of convergence you start with, while Slutsky's theorem specifically transforms a convergence-in-distribution input combined with a convergence-in-probability-to-constant input into a convergence-in-distribution output.

The solid arrows show implications between convergence modes (strongest at top-left to weakest at bottom-right). The dashed green arrows indicate that the continuous mapping theorem preserves each mode. The dashed yellow box shows Slutsky's theorem combining a convergence-in-distribution input with a convergence-in-probability-to-constant input.

A critical detail visible in the diagram is that Slutsky's theorem requires Y_n to converge in probability to a constant, not merely to another random variable. If Y_n were converging in distribution to a non-degenerate random variable Y, we could not in general conclude that X_n + Y_n →^d X + Y without additional information about the joint distribution. The continuous mapping theorem, by contrast, applies to a single sequence and a deterministic transformation g, which is why it works uniformly across all convergence modes.

SECTION 4

Mathematical Framework

Slutsky's Theorem — Formal Statement

SLUTSKY'S THEOREM

If Xₙ →ᵈ X and Yₙ →ᵖ c (c ∈ ℝ), then: (i) Xₙ + Yₙ →ᵈ X + c (ii) Xₙ · Yₙ →ᵈ c · X (iii) Xₙ / Yₙ →ᵈ X / c provided c ≠ 0

Xₙ and Yₙ are sequences of random variables defined on the same probability space. The constant c is deterministic. Convergence in probability to a constant is equivalent to convergence in distribution to a degenerate random variable at c.

The proof relies on the characterization of convergence in distribution via characteristic functions or the Portmanteau lemma. Because Y_n →^p c, the pair (X_n, Y_n) converges jointly in distribution to (X, c), which is a degenerate joint distribution. The continuous function h(x, y) = x + y (or x · y) then preserves this convergence via the continuous mapping theorem, making Slutsky's theorem a corollary of the CMT when viewed in the right framework.

Continuous Mapping Theorem — Formal Statement

CONTINUOUS MAPPING THEOREM

Let g : ℝᵏ → ℝᵐ be continuous except possibly on a set C with P(X ∈ C) = 0. Then: Xₙ →ᵈ X ⟹ g(Xₙ) →ᵈ g(X) Xₙ →ᵖ X ⟹ g(Xₙ) →ᵖ g(X) Xₙ →ᵃ·ˢ· X ⟹ g(Xₙ) →ᵃ·ˢ· g(X)

g need not be continuous everywhere—it suffices that the set of discontinuities has probability zero under the distribution of the limit X. This allows, for example, the absolute value function |·| or the indicator function 𝟙_(a,∞) when P(X = a) = 0.

The almost-sure case follows immediately from the definition: if X_n(ω) → X(ω) for ω in a set of measure 1, and g is continuous at X(ω) for almost all such ω, then g(X_n(ω)) → g(X(ω)) by the topological definition of continuity. The convergence-in-probability case uses a subsequence argument: every subsequence of g(X_n) has a further sub-subsequence converging almost surely, which then converges by the a.s. case. The convergence-in-distribution case is the most delicate and proceeds via the Portmanteau theorem: for any closed set F in ℝ^m, g⁻¹(F) is closed (up to the negligible discontinuity set), so lim sup P(g(X_n) ∈ F) ≤ P(g(X) ∈ F).

CHARACTERISTIC FUNCTION PERSPECTIVE

φ_{g(Xₙ)}(t) = E[e^{it·g(Xₙ)}] → E[e^{it·g(X)}] = φ_{g(X)}(t)

When g is continuous, the composition e^it·g(·) is bounded and continuous, so Lévy's continuity theorem guarantees that pointwise convergence of characteristic functions implies convergence in distribution.

SECTION 5

Key Applications & Classification of Uses

The power of these two theorems becomes apparent when you realize that nearly every asymptotic result in statistics involves composing simpler convergence statements. The table below classifies the most common application patterns, each of which reduces a seemingly complex derivation to a one- or two-line argument once the right theorem is invoked.

Five canonical application patterns: the CLT + WLLN combination for t-statistics (Slutsky), variance-stabilizing transforms (CMT), chi-squared derivation (CMT), Wald test statistics (Slutsky), and the Delta Method (CMT + Taylor expansion).

The most ubiquitous pattern is the derivation of the asymptotic distribution of the t-statistic. The Central Limit Theorem gives √n(X̄_n − μ)/σ →^d N(0,1), and the weak law of large numbers gives S_n →^p σ (a constant). Slutsky's theorem with the ratio form immediately yields √n(X̄_n − μ)/S_n →^d N(0,1). This one-line argument replaces what would otherwise require a careful joint-distribution calculation.

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When Slutsky Fails

If both X_n →^d X and Y_n →^d Y (with Y non-degenerate), we generally cannot determine the distribution of X_n + Y_n without knowing the joint convergence. Convergence in distribution is a statement about marginals, not joints.

SECTION 6

Worked Example — Asymptotic Distribution of a Ratio Estimator

Suppose X₁, X₂, …, X_n are i.i.d. with mean μ = 5 and variance σ² = 4. We want the asymptotic distribution of T_n = √n(X̄_n − μ) / S_n, and then the asymptotic distribution of T_n², where S_n is the sample standard deviation.

Deriving the Asymptotic Distribution of Tₙ and Tₙ²

1
Step 1 — Apply the Central Limit Theorem

By the CLT, √n(X̄_n − μ)/σ →^d Z where Z ~ N(0,1). Since μ = 5 and σ = 2, we have √n(X̄_n − 5)/2 →^d N(0,1).

Z_n = √n(X̄_n − 5)/2 →^d N(0,1)
2
Step 2 — Apply the Weak Law of Large Numbers

By the WLLN (or the consistency of the sample variance), S_n² →^p σ² = 4. Since g(x) = √x is continuous on (0, ∞) and σ² > 0, the continuous mapping theorem gives S_n = √(S_n²) →^p σ = 2.

S_n →^p 2 (a constant)
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Step 3 — Apply Slutsky's Theorem (ratio form)

We write T_n = √n(X̄_n − μ)/S_n = [√n(X̄_n − μ)/σ] × [σ/S_n]. The first factor converges in distribution to N(0,1) by Step 1. The second factor σ/S_n = 2/S_n →^p 1 by Step 2 and the CMT (since f(x) = 2/x is continuous at x = 2). By Slutsky's theorem (product form), T_n →^d 1 × N(0,1) = N(0,1).

T_n = √n(X̄_n − μ)/S_n →^d N(0,1)
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Step 4 — Apply the Continuous Mapping Theorem to get Tₙ²

Define g(t) = t². This function is continuous on all of ℝ. By the continuous mapping theorem applied to the result of Step 3, g(T_n) = T_n² →^d Z² where Z ~ N(0,1). Since Z² ~ χ²(1), we conclude that T_n² converges in distribution to a chi-squared distribution with one degree of freedom.

T_n² →^d χ²(1)

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Observe the Workflow

This example demonstrates the standard asymptotic recipe: (1) use the CLT to establish convergence in distribution, (2) use the LLN to establish convergence in probability to a constant, (3) apply Slutsky to combine them, and (4) apply the CMT to transform the result. Mastering this four-step pattern will let you handle the vast majority of asymptotic distribution problems.

SECTION 7

Comparing Slutsky's Theorem & the Continuous Mapping Theorem

Although Slutsky's theorem can be viewed as a special case of the continuous mapping theorem (as noted in Section 4), they serve distinct roles in practice and are stated with different hypotheses. The following table clarifies the similarities and differences so you can choose the right tool in each situation.

Feature	Slutsky's Theorem	Continuous Mapping Theorem
Input sequences	Two sequences: Xₙ →ᵈ X and Yₙ →ᵖ c (constant)	One sequence: Xₙ → X in any mode
Operation	Arithmetic: addition, multiplication, division	Any continuous (or a.e.-continuous) function g
Output mode	Convergence in distribution	Same mode as the input
Key restriction	One operand must converge to a constant, not a random variable	g must be continuous at least on a set of full probability under the limit
Typical use case	Replacing unknown parameters with consistent estimators (e.g., σ by Sₙ)	Transforming a convergent sequence (e.g., squaring, exponentiating, log)
Multivariate extension	Xₙ can be a vector in ℝᵏ; Yₙ can be a matrix converging to a constant matrix	Extends naturally to g : ℝᵏ → ℝᵐ

Feature comparison of the two theorems

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KEY TAKEAWAY

Think of the two theorems as complementary tools in a toolkit. The continuous mapping theorem is like a universal adapter—it lets you transform the output of any convergence result through a smooth function. Slutsky's theorem is like a calibration tool—it lets you swap in a consistent estimator for a parameter without changing the limiting distribution. In research, you will often use both in sequence: CMT to transform, Slutsky to plug in estimated nuisance parameters.

SECTION 8

Connections to Advanced Theory

Slutsky's theorem and the continuous mapping theorem serve as gateways to more sophisticated asymptotic tools. In advanced courses you will encounter generalizations that relax hypotheses, extend to function spaces, or incorporate rates of convergence. The table below provides a map from the introductory versions you have learned here to their more general counterparts.

Introductory Result	Advanced Generalization	Key Extension
CMT for random variables in ℝᵏ	CMT for random elements in metric spaces	Xₙ can be a random function in C[0,1] or D[0,1]; g : C[0,1] → ℝ
Slutsky's theorem (Yₙ →ᵖ c)	Joint convergence + CMT	If (Xₙ, Yₙ) →ᵈ (X, Y) jointly, then h(Xₙ, Yₙ) →ᵈ h(X, Y) for continuous h
CMT + Delta Method	Second-order Delta Method	When g'(θ) = 0, the rate changes to n (not √n) and the limit is a scaled χ² distribution
Convergence in distribution in ℝ	Weak convergence in (D[0,1], J₁)	Functional CLT (Donsker's theorem); process-level convergence where CMT applies to functionals like sup, integral, etc.

From introductory to advanced asymptotic theory

Perhaps the most important advanced application is the Delta Method, which is essentially a first-order Taylor expansion combined with the continuous mapping theorem. If √n(θ̂_n − θ) →^d N(0, σ²) and g is differentiable at θ with g'(θ) ≠ 0, then √n(g(θ̂_n) − g(θ)) →^d N(0, [g'(θ)]²σ²). This result is used ubiquitously in maximum likelihood theory, biostatistics, and econometrics to construct confidence intervals for transformed parameters.

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Looking Ahead

In functional data analysis and empirical process theory, the CMT operates on entire stochastic processes rather than finite-dimensional vectors. For example, once Donsker's theorem establishes that the empirical process converges weakly to a Brownian bridge, the CMT implies that the Kolmogorov–Smirnov statistic (a supremum functional) converges in distribution to the supremum of the Brownian bridge.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why Slutsky's theorem requires that Y_n converge in probability to a constant, rather than to an arbitrary random variable Y. Provide a concrete counterexample showing that the conclusion can fail when Y is non-degenerate.

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Convergence in distribution specifies only marginal CDFs, not the joint distribution. If X_n →ᵈ X and Y_n →ᵈ Y, the distribution of X_n + Y_n depends on the dependence structure. For example, let X_n = Z ~ N(0,1) for all n. Set Y_n = Z (same as X_n); then X_n + Y_n = 2Z ~ N(0,4). Alternatively, set Y_n = −Z; then X_n + Y_n = 0 a.s. Both Y_n choices have the same marginal N(0,1), but the sums have completely different distributions. When Y_n →ᵖ c, the joint distribution becomes degenerate in the Y-coordinate, resolving this ambiguity.

Question 2: BASIC CALCULATION

Let X₁, …, X_n be i.i.d. with E[X₁] = 3 and Var(X₁) = 9. Find the asymptotic distribution of √n(X̄_n − 3)/S_n, where S_n is the sample standard deviation.

Show answer

By the CLT, √n(X̄_n − 3)/3 →ᵈ N(0,1). By the WLLN and the CMT (since √· is continuous on (0,∞)), S_n →ᵖ 3. Writing the statistic as [√n(X̄_n − 3)/3] × (3/S_n), the first factor converges in distribution to N(0,1) and 3/S_n →ᵖ 1. By Slutsky's theorem (product form), √n(X̄_n − 3)/S_n →ᵈ N(0,1).

Question 3: INTERMEDIATE

Let X₁, …, X_n be i.i.d. Bernoulli(p) with p ∈ (0,1). Use the CLT, Slutsky's theorem, and the continuous mapping theorem to find the asymptotic distribution of n(p̂_n − p)² / [p̂_n(1 − p̂_n)], where p̂_n = X̄_n.

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By the CLT, √n(p̂_n − p)/√(p(1−p)) →ᵈ N(0,1). By the WLLN, p̂_n →ᵖ p. The function g(x) = x(1−x) is continuous, so by the CMT, p̂_n(1−p̂_n) →ᵖ p(1−p). Now write the statistic as [√n(p̂_n − p)/√(p(1−p))]² × [p(1−p)/(p̂_n(1−p̂_n))]. The first factor is Z_n² where Z_n →ᵈ N(0,1), so by the CMT (g(z)=z²), Z_n² →ᵈ χ²(1). The second factor converges in probability to 1 by the CMT applied to the ratio. By Slutsky (product), the full statistic →ᵈ χ²(1). This is the one-sample Wald (or score) test statistic for a proportion.

Question 4: APPLIED

In a clinical trial, suppose n patients yield i.i.d. survival times with unknown mean μ and finite variance σ². A researcher constructs the log-transformed confidence interval by first computing √n(X̄_n − μ)/S_n and then applying g(t) = e^t. What is the asymptotic distribution of exp(√n(X̄_n − μ)/S_n)? Is this distribution normal? Explain using the theorems from this lesson.

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By the CLT + Slutsky (as in the worked example), T_n = √n(X̄_n − μ)/S_n →ᵈ N(0,1). Since g(t) = eᵗ is continuous on ℝ, the continuous mapping theorem gives exp(T_n) →ᵈ e^Z where Z ~ N(0,1). The distribution of e^Z is a log-normal distribution with parameters 0 and 1—it is decidedly not normal. Its PDF is f(x) = (1/(x√(2π))) exp(−(ln x)²/2) for x > 0. This illustrates how the CMT can map a normal limit to a non-normal one; the theorem preserves the mode of convergence but not the family of distributions.

Question 5: CRITICAL THINKING

Consider a sequence X_n →ᵈ X where X ~ Uniform(0,1). Define g(x) = 𝟙(x ≤ 0.5), the indicator function. Does g(X_n) →ᵈ g(X)? What if instead X ~ Uniform(0,1) is replaced by X ~ Uniform(0,1) conditioned so that P(X = 0.5) = 0.3 (i.e., X has a point mass at 0.5)? Justify your answers using the precise hypothesis of the continuous mapping theorem.

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The function g(x) = 𝟙(x ≤ 0.5) is discontinuous at exactly x = 0.5. The CMT allows discontinuities on a set C provided P(X ∈ C) = 0. In the first scenario, X ~ Uniform(0,1), so P(X = 0.5) = 0. The discontinuity set C = {0.5} has probability zero, so the CMT applies and g(X_n) →ᵈ g(X) ~ Bernoulli(0.5). In the second scenario, P(X = 0.5) = 0.3 > 0, so the discontinuity set has positive probability and the CMT does not directly apply. Indeed, the conclusion can fail: the behavior of g(X_n) near the discontinuity depends on whether X_n approaches 0.5 from the left or right, which convergence in distribution does not specify. This example demonstrates that the 'almost everywhere continuity' condition in the CMT is not merely a technical convenience—it is essential.

SUMMARY

Summary & Review

Summary

Slutsky's theorem states that if Xₙ converges in distribution to X and Yₙ converges in probability to a constant c, then their sum, product, and ratio converge in distribution to X + c, cX, and X/c respectively. The requirement that the limit of Yₙ be a constant (degenerate random variable) is essential because convergence in distribution captures only marginal behavior, not joint dependence. This theorem is the workhorse for replacing unknown parameters with consistent estimators in asymptotic arguments—for instance, substituting the sample standard deviation for the population standard deviation in a z-statistic to produce a t-statistic.

The continuous mapping theorem (CMT) asserts that applying a continuous function g to a convergent sequence preserves the mode of convergence—whether almost sure, in probability, or in distribution. Crucially, g need only be continuous on a set whose complement has probability zero under the limit distribution. Together, these two theorems form the algebraic engine of asymptotic statistics, enabling the derivation of limiting distributions for test statistics, confidence intervals, and estimators through a modular, composable framework that extends naturally to the Delta Method and functional weak convergence.