# Andersen theorem

A result in the theory of fluctuations in random walks (cf. Random walk). Let $( X _ {n} ) _ {1} ^ \infty$ be independent random variables with the same distribution (cf. Random variable), and let $S _ {0} = 0$, $S _ {k} = X _ {1} + \dots + X _ {k}$, $k \in \mathbf N$. Define

$$M _ {n} = \max ( S _ {0} \dots S _ {n} ) , m _ {n} = { \mathop{\rm min} } ( S _ {0} \dots S _ {n} ) ,$$

$$L _ {n} = { \mathop{\rm min} } \left \{ k : {k = 0 \dots n, S _ {k} = M _ {n} } \right \} ,$$

$$L _ {n} ^ \prime = \max \left \{ k : {k = 0 \dots n, S _ {k} = m _ {n} } \right \} ,$$

$$N _ {n} = \sum _ {k = 1 } ^ { n } 1 \{ S _ {k} > 0 \} .$$

Then (equivalence principle) for each $n \in \mathbf N$ the pairs $( N _ {n} ,S _ {n} )$, $( L _ {n} ,S _ {n} )$ and $( n - L _ {n} ^ \prime ,S _ {n} )$ have the same distribution; in particular, $N _ {n}$, $L _ {n}$ and $n - L _ {n} ^ \prime$ have the same distribution. As a consequence one has

$${\mathsf P} \{ N _ {n} = k \} = {\mathsf P} \{ N _ {k} = k \} {\mathsf P} \{ N _ {n - k } = 0 \} ,$$

$$k = 1 \dots n .$$

These results were first proved by E. Sparre Andersen [a1], [a2], [a3]. They connect the arcsine law for random walks to the arcsine law in renewal theory.

Nowadays there are brief proofs based on combinatorial properties of non-random sequences [a6], [a7]. The results can be generalized to random vectors with symmetric distributions [a2]. A comprehensive account for integer-valued random variables can be found in [a8]; a concise overview is given in [a4]. Related combinatorial results are discussed in [a5].

#### References

 [a1] E. Sparre Andersen, "On the number of positive sums of random variables" Skand. Aktuarietikskr., 32 (1949) pp. 27–36 [a2] E. Sparre Andersen, "On sums of symmetrically dependent random variables" Skand. Aktuarietikskr., 36 (1953) pp. 123–138 [a3] E. Sparre Andersen, "On the fluctuations of sums of random variables" Math. Scand., 1 (1953) pp. 263–285 (Also: 2 (1954), 195–223) [a4] N.H. Bingham, C.M. Goldie, J.L. Teugels, "Regular variation", Encycl. Math. Appl., 27, Cambridge Univ. Press (1989) (Edition: Second) [a5] N.G. de Bruijn, "Some algorithms for ordering a sequence of objects, with application to E. Sparre Andersen's principle of equivalence in mathematical statistics" Indagationes Mathematicae, 34 : 1 (1972) pp. 1–10 [a6] W. Feller, "An introduction to probability theory and its applications", 2, Springer (1976) (Edition: Second) [a7] A.W. Joseph, "An elementary proof of the principle of equivalence" J. London Math. Soc. (2), 3 (1971) pp. 101–102 [a8] F. Spitzer, "Principles of random walk", Springer (1976) (Edition: Second)
How to Cite This Entry:
Andersen theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Andersen_theorem&oldid=45184
This article was adapted from an original article by F.W. Steutel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article