It is well known that the variance of the sum of $x_1,...,x_N$ random variables is the sum of their variances plus twice their covariances:

$\text{Var} \displaystyle\sum_{i=1}^{N}x_i =\displaystyle\sum_{i=1}^{N}\text{Var} \{x_i\} + 2\sum_{i\neq j}\text{Cov}\{x_i,x_j\}$ (1)

The expression above can be found in many text books. However, as $N \rightarrow \infty$, couldn't we replace the summations by integrals and express the equation above using integral notation? Something like the equation below, for instance:

$\text{Var} \displaystyle\int x \text{d} x =\displaystyle\int\text{Var} \{x\} \text{d} x + 2\int\int\text{Cov}\{x_i,x_j\}\text{d} x_i \text{d} x_j$ (2)

Is there any equation out there for the variance of the sum of random variables that use integrals? I have never seen any textbook/paper discussing if the summation should become integrals in the variace of the sum formula when $N \rightarrow \infty$, or if we could express such a classical expression as (1) using integrals as (2). Is it possible to use integral notation? Does any reference exist on this topic?

  • $\begingroup$ What does $\lim_{N\to \infty} \sum_{i=1}^{N}x_i$ look like? $\endgroup$ – user158565 Oct 27 '18 at 23:09
  • $\begingroup$ Hello, thanks for your reply. This limit might converge, diverge or be indefinitely divergent depending on how $x_i$ looks like.. right? $\endgroup$ – antamoeba Oct 28 '18 at 13:57

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