Spitzer's formula

Last updated August 27, 2021

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

\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

{\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]

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References

1 2 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 .
↑ 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.
1 2 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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[spitzer-1] 1 2 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.

[wendel-3] 1 2 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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Spitzer's formula

Contents

Statement of theorem

Proof

Related Research Articles

References