Slutsky’s Theorem: Definition

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Slutsky’s theorem is used to explore convergence in probability distributions. It tells us that if a sequence of random vectors converges in distribution and another sequence converges in probability to a constant, those sequences are jointly convergent in distribution. Basically, it allows you to use convergence results proved for one sequence for other closely related sequences.

Slutsky’s has no real practical applications; Its use is mostly limited to theoretical mathematical statistics (specifically, asymptotic theory). For example, it extends the usefulness of the Central Limit Theorem. Other uses include:

• Explore convergence in functions of random variables.
• Highlight critical properties of converging random variables.
• Calculate the convergence of any continuous function of a set of statistics, providing that one set of those statistics converges (Davidson, 1994).

Formal Definition of Slutsky’s Theorem

More formally, Manoukian (1986) defines Slutsky’s theorem as follows:

If X_i be a random variable sequence that converges to a random variable X with a distribution function F(x) and if Y_i is a random variable sequence that converges to a probability of constant c. Then:

1. X_i + Y_i is distributed asymptotically as X + c.
2. X_i Y_i is distributed asymptotically as Xc.
3. X_i / Y_i is distributed asymptotically as X / c for c ≠ 0.

The theorem can also be written more succinctly as (from Proschan & Shaw, 2016):

Suppose that Xn→DX, An↠pA, and Bn↠pB,
where A and B are constants. Then AnXn+Bn↠DAX+B.

Simple Example

First, we need to define a couple of functions, g and h. The family of functions g_i, is defined as:

• g₁({x_n}) = {x_n}
• g₂({x_n}) = {2x_n}
• g₃({x_n}) = {3x_n}
• …
• g_k({x_n}) = {kx_n}

g_i converges in probability to a constant c = μ.
And h is defined, in terms of g, as:
h (g₁, g₂, g₃…g_k) =

With reference to these two functions, Slutsky’s theorem tells us that the limit of h (g₁, g₂, g₃…g_k) as n approaches infinity is:
(k(k + 1) / 2) · μ
(Adapted from Kapadia et.al)

References:

Davidson, J. (1994). Stochastic Limit Theory: An Introduction for Econometricians.
Kapadia et. al. (2005). Mathematical Statistics With Applications.
Manoukian (1986) Mathematical Nonparametric Statistics. CRC Press.
Proschan, M. & Shaw, P. (2016). Essentials of Probability Theory for Statisticians. CRC Press.

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