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 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[3]

where

and S± denotes (|S| ± S)/2.

Proof

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

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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