# 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].

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