Slutsky's theorem

In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.{{cite book |first=Arthur S. |last=Goldberger |author-link=Arthur Goldberger |title=Econometric Theory |location=New York |publisher=Wiley |year=1964 |pages=[https://archive.org/details/econometrictheor0000gold/page/117 117]–120 |url=https://archive.org/details/econometrictheor0000gold |url-access=registration|ref=NULL }}

The theorem was named after Eugen Slutsky.{{Cite journal

| last = Slutsky | first = E. | author-link = Eugen Slutsky

| year = 1925

| title = Über stochastische Asymptoten und Grenzwerte

| language = de

| journal = Metron

| volume = 5 | issue = 3

| pages = 3–89

| jfm = 51.0380.03

| ref = NULL

}} Slutsky's theorem is also attributed to Harald Cramér.Slutsky's theorem is also called Cramér's theorem according to Remark 11.1 (page 249) of {{Cite book

| last = Gut | first = Allan

| title = Probability: a graduate course

| publisher = Springer-Verlag

| year = 2005

| isbn = 0-387-22833-0

| ref = NULL

}}

Statement

Let $X_n, Y_n$ be sequences of scalar/vector/matrix random elements.

If $X_n$ converges in distribution to a random element $X$ and $Y_n$ converges in probability to a constant $c$ , then

$X_n + Y_n \ \xrightarrow{d}\ X + c ;$
$X_nY_n \ \xrightarrow{d}\ Xc ;$
$X_n/Y_n \ \xrightarrow{d}\ X/c,$ provided that c is invertible,

where $\xrightarrow{d}$ denotes convergence in distribution.

Notes:

The requirement that Y_n converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. For example, let $X_n \sim {\rm Uniform}(0,1)$ and $Y_n = -X_n$ . The sum $X_n + Y_n = 0$ for all values of n. Moreover, $Y_n \, \xrightarrow{d} \, {\rm Uniform}(-1,0)$ , but $X_n + Y_n$ does not converge in distribution to $X + Y$ , where $X \sim {\rm Uniform}(0,1)$ , $Y \sim {\rm Uniform}(-1,0)$ , and $X$ and $Y$ are independent.See {{cite web |first=Donglin |last=Zeng |title=Large Sample Theory of Random Variables (lecture slides) |work=Advanced Probability and Statistical Inference I (BIOS 760) |url=https://www.bios.unc.edu/~dzeng/BIOS760/ChapC_Slide.pdf#page=59 |publisher=University of North Carolina at Chapel Hill |date=Fall 2018 |at=Slide 59 }}
The theorem remains valid if we replace all convergences in distribution with convergences in probability.

Proof

This theorem follows from the fact that if X_n converges in distribution to X and Y_n converges in probability to a constant c, then the joint vector (X_n, Y_n) converges in distribution to (X, c) (see here).

Next we apply the continuous mapping theorem, recognizing the functions g(x,y) = x + y, g(x,y) = xy, and g(x,y) = x y⁻¹ are continuous (for the last function to be continuous, y has to be invertible).

Slutsky's theorem

Statement

Proof

See also

References

Further reading