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 S_n = X₁ + X₂ + ... + X_n. Define R_n = max(0,S₁,S₂,...,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

$\begin{align} \phi_n(\alpha,\beta) &= \operatorname E(\exp\left[ i(\alpha R_n + \beta(R_n-S_n)\right])\\ u_n(\alpha) &= \operatorname E(\exp \left[i\alpha S_n^+\right]) \\ v_n(\beta) &= \operatorname E(\exp \left[i \beta S_n^-\right]) \end{align}$

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. 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.