Spitzer's formula

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.^[1] The formula is regarded as "a stepping stone in the theory of sums of independent random variables".^[2]

Statement of theorem

Let ${\displaystyle X_{1},X_{2},...}$ be independent and identically distributed random variables and define the partial sums ${\displaystyle S_{n}=X_{1}+X_{2}+...+X_{n}}$. Define ${\displaystyle R_{n}={\text{max}}(0,S_{1},S_{2},...S_{n})}$. Then^[3]

${\displaystyle \sum _{n=0}^{\infty }\phi _{n}(\alpha ,\beta )t^{n}=\exp \left[\sum _{n=1}^{\infty }{\frac {t^{n}}{n}}\left(u_{n}(\alpha )+v_{n}(\beta )-1\right)\right]}$

where

${\displaystyle {\begin{aligned}\phi _{n}(\alpha ,\beta )&=\operatorname {E} (\exp \left[i(\alpha R_{n}+\beta (R_{n}-S_{n})\right])\\u_{n}(\alpha )&=\operatorname {E} (\exp \left[i\alpha S_{n}^{+}\right])\\v_{n}(\beta )&=\operatorname {E} (\exp \left[i\beta S_{n}^{-}\right])\end{aligned}}}$

and S^± denotes (|S| ± S)/2.

Proof

Two proofs are known, due to Spitzer^[1] and Wendel.^[3]