# Slutsky's theorem

The set of Slutsky or the Slutsky theorem developed by Jewgeni Slutskys (E. Slutsky) is a mathematical theorem in the field of probability theory , the convergence of random variables relates.

## theorem

If the sequence of random variables for towards infinity converges to the random variable in distribution and the sequences of random variables and to the values or in probability converge , then the function in distribution converges to . Short: ${\ displaystyle X_ {n} \;}$${\ displaystyle n}$${\ displaystyle X \;}$ ${\ displaystyle A_ {n} \;}$${\ displaystyle B_ {n} \;}$${\ displaystyle a}$${\ displaystyle b}$ ${\ displaystyle A_ {n} + B_ {n} X_ {n} \;}$${\ displaystyle a + bX \;}$

${\ displaystyle A_ {n} + B_ {n} X_ {n} \ {\ stackrel {D} {\ rightarrow}} \ a + bX}$

## literature

• Erich L. Lehmann: Elements of large sample theory . Springer, New York 1999, ISBN 0-387-98595-6 , pp. 70 .
• Harald Cramér: Mathematical Methods of Statistics . Princeton University Press, Princeton 1946.