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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".

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

${\displaystyle \sum _{n=0}^{\infty }\phi _{n}(\alpha ,\beta )t^{n}=\exp \left}$

where

${\displaystyle {\begin{aligned}\phi _{n}(\alpha ,\beta )&=\operatorname {E} (\exp \left)\\u_{n}(\alpha )&=\operatorname {E} (\exp \left)\\v_{n}(\beta )&=\operatorname {E} (\exp \left)\end{aligned}}}$

and S denotes (|S| ± S)/2.

Proof

Two proofs are known, due to Spitzer and Wendel.

References

1. ^ Spitzer, F. (1956). "A combinatorial lemma and its application to probability theory". Transactions of the American Mathematical Society. 82 (2): 323–339. doi:10.1090/S0002-9947-1956-0079851-X.
2. Ebrahimi-Fard, K.; Guo, L.; Kreimer, D. (2004). "Spitzer's identity and the algebraic Birkhoff decomposition in pQFT". Journal of Physics A: Mathematical and General. 37 (45): 11037. arXiv:hep-th/0407082. Bibcode:2004JPhA...3711037E. doi:10.1088/0305-4470/37/45/020.
3. ^ Wendel, James G. (1958). "Spitzer's formula: A short proof". Proceedings of the American Mathematical Society. 9 (6): 905–908. doi:10.1090/S0002-9939-1958-0103531-2. MR 0103531.

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