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)