Slutsky's theorem

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.[1] The theorem was named after Eugen Slutsky.[2] Slutsky's theorem is also attributed to Harald Cramér.[3] Contents 1 Statement 2 Proof 3 See also 4 References 5 Further reading Statement Let {displaystyle X_{n},Y_{n}} be sequences of scalar/vector/matrix random elements. If {displaystyle X_{n}} converges in distribution to a random element {displaystyle X} and {displaystyle Y_{n}} converges in probability to a constant {displaystyle c} , then {displaystyle X_{n}+Y_{n} {xrightarrow {d}} X+c;} {displaystyle X_{n}Y_{n} xrightarrow {d} Xc;} {displaystyle X_{n}/Y_{n} {xrightarrow {d}} X/c,}   provided that c is invertible, where {displaystyle {xrightarrow {d}}} denotes convergence in distribution.

Notes: The requirement that Yn converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 {displaystyle X_{n}sim {rm {Uniform}}(0,1)} and {displaystyle Y_{n}=-X_{n}} . The sum {displaystyle X_{n}+Y_{n}=0} for all values of n. Moreover, {displaystyle Y_{n},xrightarrow {d} ,{rm {Uniform}}(-1,0)} , but {displaystyle X_{n}+Y_{n}} does not converge in distribution to {displaystyle X+Y} , where {displaystyle Xsim {rm {Uniform}}(0,1)} , {displaystyle Ysim {rm {Uniform}}(-1,0)} , and {displaystyle X} and {displaystyle Y} are independent.[4] The theorem remains valid if we replace all convergences in distribution with convergences in probability. Proof This theorem follows from the fact that if Xn converges in distribution to X and Yn converges in probability to a constant c, then the joint vector (Xn, Yn) 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).

See also Convergence of random variables References ^ Goldberger, Arthur S. (1964). Econometric Theory. New York: Wiley. pp. 117–120. ^ Slutsky, E. (1925). "Über stochastische Asymptoten und Grenzwerte". Metron (in German). 5 (3): 3–89. JFM 51.0380.03. ^ Slutsky's theorem is also called Cramér's theorem according to Remark 11.1 (page 249) of Gut, Allan (2005). Probability: a graduate course. Springer-Verlag. ISBN 0-387-22833-0. ^ See Zeng, Donglin (Fall 2018). "Large Sample Theory of Random Variables (lecture slides)" (PDF). Advanced Probability and Statistical Inference I (BIOS 760). University of North Carolina at Chapel Hill. Slide 59. Further reading Casella, George; Berger, Roger L. (2001). Statistical Inference. Pacific Grove: Duxbury. pp. 240–245. ISBN 0-534-24312-6. Grimmett, G.; Stirzaker, D. (2001). Probability and Random Processes (3rd ed.). Oxford. Hayashi, Fumio (2000). Econometrics. Princeton University Press. pp. 92–93. ISBN 0-691-01018-8. Categories: Asymptotic theory (statistics)Probability theoremsTheorems in statistics

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