Slutsky's theorem

Slutsky's theorem is a result in probability theory that describes how convergence in distribution interacts with convergence in probability to a constant. It permits the limiting distribution of a statistic to remain unchanged when an unknown deterministic quantity is replaced by a consistent random approximation. The theorem consequently underlies many constructions in asymptotic statistics, including standardized estimators and statistics containing estimated nuisance parameters.

The theorem takes its name from Eugen Slutsky, who established its scalar form while studying stochastic limits during the 1920s. A contemporaneous random-vector formulation was developed by You Watanabe, whose treatment expressed the result through continuous transformations of jointly convergent quantities. The modern theorem combines these formulations within the general framework of weak convergence.

Statement

Let (X_n) and (Y_n) be sequences of random variables defined on a common probability space. Suppose that

[ X_n \xrightarrow{d} X ]

and

[ Y_n \xrightarrow{p} c, ]

where (\xrightarrow{d}) denotes convergence in distribution, (\xrightarrow{p}) denotes convergence in probability, and (c) is a constant. Slutsky's theorem states that

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

[ X_nY_n \xrightarrow{d} cX. ]

If (c\neq 0), it also gives

[ \frac{X_n}{Y_n}\xrightarrow{d}\frac{X}{c}. ]

No independence assumption between (X_n) and (Y_n) is required. The decisive condition is that the limiting value of (Y_n) is nonrandom. Convergence in probability to that constant prevents the residual randomness of (Y_n) from contributing to the limiting distribution.

For random vectors, let (X_n) take values in (\mathbb R^k), and let (Y_n) take values in (\mathbb R^m). Under the assumptions

[ X_n\xrightarrow{d}X \qquad\text{and}\qquad Y_n\xrightarrow{p}c, ]

the pair satisfies

[ (X_n,Y_n)\xrightarrow{d}(X,c). ]

Consequently, if (g:\mathbb R^{k+m}\to\mathbb R^r) is continuous at every point ((x,c)) belonging to a set on which (X) is concentrated, then the continuous mapping theorem yields

[ g(X_n,Y_n)\xrightarrow{d}g(X,c). ]

This formulation includes addition, multiplication, and division as particular continuous transformations. Division requires the limiting denominator to be nonzero because the quotient map is discontinuous where its denominator vanishes.

Mathematical basis

The central assertion of the theorem is the joint-convergence relation

[ (X_n,Y_n)\xrightarrow{d}(X,c). ]

Convergence in probability of (Y_n) implies convergence in distribution to the degenerate random variable equal to (c). Convergence of the two marginal sequences alone does not ordinarily establish joint convergence, since their dependence may affect the joint law. The constant character of the second limit removes this obstruction: every subsequential joint limit must have first marginal distributed as (X) and second coordinate equal to (c) almost surely.

A formulation using bounded continuous functions makes this mechanism explicit. For every bounded continuous function (f),

[ \mathbb E[f(X_n,Y_n)]

\mathbb E[f(X_n,c)] \longrightarrow 0, ]

while convergence in distribution of (X_n) gives

[ \mathbb E[f(X_n,c)] \longrightarrow \mathbb E[f(X,c)]. ]

Together, these relations characterize weak convergence of the pair to ((X,c)). Applying an appropriate continuous function to the pair then produces each familiar algebraic form of the theorem.

The same reasoning extends beyond Euclidean spaces. If (X_n) takes values in a metric space, (Y_n) converges in probability to a fixed point in another metric space, and the relevant transformation is continuous on the support of the limiting pair, the corresponding transformed variables converge in distribution. This version situates Slutsky's theorem within the broader theory of probability measures on topological spaces.

Historical development

Slutsky presented the original result in his 1925 work on stochastic asymptotes and limiting values. His formulation addressed the behavior of algebraic combinations in which one random component possessed a limiting distribution while another approached a constant. The result connected earlier work on limit laws with the emerging statistical practice of replacing population quantities by sample-based estimates.

Watanabe's 1926 analysis reformulated the argument for finite-dimensional random vectors and identified joint convergence to ((X,c)) as the common basis of the algebraic cases. This treatment separated the probabilistic convergence statement from the continuity of the final transformation, thereby anticipating the later continuous-mapping formulation.

In subsequent expositions, Harald Cramér incorporated the theorem into the systematic theory of statistical limit distributions and emphasized its use in replacing deterministic normalizing constants with consistent estimators. Patrick Billingsley later expressed the result through weak convergence and mapping theorems, which became the standard measure-theoretic presentation.

The designation “Cramér–Slutsky theorem” occurs in parts of the statistical literature, particularly where the result is presented alongside Cramér's treatment of asymptotic distributions. “Slutsky's theorem” remains the more common name in probability and mathematical statistics.

Statistical interpretation

A principal application concerns a statistic whose limiting distribution is known after normalization by a population parameter. Suppose that

[ \sqrt n,(T_n-\theta)\xrightarrow{d}N(0,\sigma^2), ]

where (N(0,\sigma^2)) is a normal distribution and (\sigma>0). If an estimator (\widehat{\sigma}_n) satisfies

[ \widehat{\sigma}_n\xrightarrow{p}\sigma, ]

then Slutsky's theorem gives

[ \frac{\sqrt n,(T_n-\theta)} {\widehat{\sigma}_n} \xrightarrow{d}N(0,1). ]

Thus the limiting distribution is preserved when the unknown scale is replaced by a consistent estimator. This substitution is the mathematical basis of many asymptotically pivotal statistics.

The theorem also connects consistency with limiting distribution theory. If

[ T_n\xrightarrow{p}\theta ]

and

[ \sqrt n,(S_n-\eta)\xrightarrow{d}Z, ]

then products or smooth combinations involving (T_n) can often be reduced to transformations of (Z) and the constant (\theta). The resulting limit depends on the deterministic probability limit of the consistent component rather than on its finite-sample distribution.

This role differs from that of the delta method. Slutsky's theorem replaces a convergent random component by its constant limit inside a continuous transformation. The delta method instead determines the first-order distributional effect of applying a differentiable transformation to a fluctuating estimator. Statistical derivations frequently combine the two results because one controls estimated normalizations while the other controls nonlinear transformations.

Equivalent consequences

A frequently used consequence concerns asymptotically equivalent sequences. If

[ X_n-Y_n\xrightarrow{p}0 ]

and

[ X_n\xrightarrow{d}X, ]

then

[ Y_n\xrightarrow{d}X. ]

Indeed, (Y_n=X_n+(Y_n-X_n)), and the second term converges in probability to zero. The theorem therefore formalizes the statement that a perturbation vanishing in probability does not alter a limiting distribution.

A related multiplicative form applies when

[ \frac{Y_n}{X_n}\xrightarrow{p}1 ]

and the relevant denominators are defined with probability approaching one. Under suitable nonvanishing conditions on the limit, (X_n) and (Y_n) then possess the same normalized asymptotic behavior. This relation is commonly expressed using asymptotic equivalence.

The conclusion changes when the second component converges to a nondegenerate random variable. If

[ X_n\xrightarrow{d}X \qquad\text{and}\qquad Y_n\xrightarrow{d}Y, ]

the marginal limits do not determine the limiting distribution of (X_n+Y_n) or (X_nY_n). A joint-convergence assumption is then necessary, because different dependence structures can share the same marginal distributions while producing different distributions for their sums or products.