A.3 Iterative mean

The mean value of a distribution {x_i} can also be computed iteratively if the values x_i are drawn one-by-one. Let $\left\langle\vphantom{ x }\right.$ x $\left.\vphantom{ x }\right\rangle_{t}^{}$ be the average over the first t data points. We observe that

$\displaystyle \left\langle\vphantom{ x }\right.$ x $\displaystyle \left.\vphantom{ x }\right\rangle_{t}^{}$	=	$\displaystyle {\frac{{1}}{{t}}}$ $\displaystyle \sum_{{i=1}}^{t}$ x_i
	=	$\displaystyle {\frac{{1}}{{t}}}$ $\displaystyle \sum_{{i=1}}^{{t-1}}$ x_i + $\displaystyle {\frac{{1}}{{t}}}$ x_t
	=	$\displaystyle {\frac{{t-1}}{{t}}}$ $\displaystyle {\frac{{1}}{{t-1}}}$ $\displaystyle \sum_{{i=1}}^{{t-1}}$ x_i + $\displaystyle {\frac{{1}}{{t}}}$ x_t
	=	$\displaystyle {\frac{{t-1}}{{t}}}$ $\displaystyle \left\langle\vphantom{ x }\right.$ x $\displaystyle \left.\vphantom{ x }\right\rangle_{{t-1}}^{}$ + $\displaystyle {\frac{{1}}{{t}}}$ x_t .	(A.13)

c(t + 1) = c(t) + $\displaystyle {\frac{{1}}{{t+1}}}$ $\displaystyle \left(\vphantom{x - c(t)}\right.$ x - c(t) $\displaystyle \left.\vphantom{x - c(t)}\right)$ .

(A.14)