Bienaymé's identity

In probability theory, the general form of Bienaymé's identity, named for Irénée-Jules Bienaymé, states that

Estimated variance of the cumulative sum of iid normally distributed random variables (which could represent a gaussian random walk approximating a Wiener process). The sample variance is computed over 300 realizations of the corresponding random process.

$${\begin{aligned}\operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)&=\sum _{i=1}^{n}\operatorname {Var} (X_{i})+2\sum _{i,j=1 \atop i<j}^{n}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i,j=1}^{n}\operatorname {Cov} (X_{i},X_{j}).\end{aligned}}$$

This can be simplified if $X_{1},\ldots ,X_{n}$ are pairwise independent or just uncorrelated, integrable random variables, each with finite second moment. This simplification gives:

$$\operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{k=1}^{n}\operatorname {Var} (X_{k}).$$

The above expression is sometimes referred to as Bienaymé's formula. Bienaymé's identity may be used in proving certain variants of the law of large numbers.

See also

• Variance
• Propagation of error
• Markov chain central limit theorem
• Panjer recursion
• Inverse-variance weighting
• Donsker's theorem
• Paired difference test