Slutsky's Theorem

In statistics, an estimator is often built from two pieces at once: one piece that settles down to a genuinely random limit (like a sample mean, which by the central limit theorem converges to a normal distribution), and another piece that settles down to a fixed, non-random number (like a sample variance estimate, which converges to the true variance). Slutsky's theorem tells you exactly how to combine these two different kinds of limits and still get a sensible answer.

Formally, if $X_n \xrightarrow{d} X$ (converges in distribution to a random variable $X$) and $Y_n \xrightarrow{p} c$ (converges in probability to a constant $c$), then:

$$X_n + Y_n \xrightarrow{d} X + c, \qquad X_n Y_n \xrightarrow{d} cX, \qquad \frac{X_n}{Y_n} \xrightarrow{d} \frac{X}{c} \; (c \neq 0).$$

In plain English: once one of the two pieces has stopped being random and locked onto a fixed number, you can treat it like an ordinary constant when combining it with the still-random piece — the randomness of $X_n$ passes through additions, multiplications and divisions unchanged, just scaled or shifted by $c$.

Worked example. Say $\bar{X}_n \xrightarrow{d} N(0, 1)$ (a standardized sample mean) and $s_n \xrightarrow{p} 2$ (a sample standard deviation converging to the true value 2). Slutsky's theorem says $\bar{X}_n / s_n \xrightarrow{d} N(0,1)/2$, i.e. a normal distribution scaled to have standard deviation $1/2$ — you don't need to track the joint randomness of $\bar X_n$ and $s_n$ together, because $s_n$'s randomness has already died out in the limit.

This is exactly the justification for the standard t-statistic construction: dividing an asymptotically normal quantity by an estimated standard error and still getting a known limiting distribution, which is what makes confidence intervals and hypothesis tests built from estimated variances valid in large samples.