Spitzer's formula
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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.[1] The formula is regarded as "a stepping stone in the theory of sums of independent random variables".[2]
Statement of theorem
Let [math]\displaystyle{ X_1,X_2,... }[/math] be independent and identically distributed random variables and define the partial sums [math]\displaystyle{ S_n=X_1 + X_2 + ... + X_n }[/math]. Define [math]\displaystyle{ R_n=\text{max}(0,S_1,S_2,...S_n) }[/math]. Then[3]
- [math]\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] }[/math]
where
- [math]\displaystyle{ \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} }[/math]
and S± denotes (|S| ± S)/2.
Proof
Two proofs are known, due to Spitzer[1] and Wendel.[3]
References
- ↑ 1.0 1.1 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.
- ↑ 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. Bibcode: 2004JPhA...3711037E.
- ↑ 3.0 3.1 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.
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