In probability theory, Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published by Frank Spitzer in 1956. The formula is regarded as "a stepping stone in the theory of sums of independent random variables".
Contents
Statement of theorem
Let X1, X2, ... be independent and identically distributed random variables and define the partial sums Sn = X1 + X2 + ... + Xn. Define Rn = max(0,S1,S2,...,Sn). Then
where
and S± denotes (|S| ± S)/2.
Proof
Two proofs are known, due to Spitzer and Wendel.
References
Spitzer's formula Wikipedia(Text) CC BY-SA