# Why is the variance of ACF of white noise 1/T

In many books, articles and comments on this website I read that the variance of the autocorrelation of a white noise process is $\frac{1}{T}$ when T is sufficiently large. Often this characteristic is used to calculate the confidence bounds in an ACF plot, where the bounds are calculated by: $b = \pm 1.96 \frac{1}{\sqrt{T}}$, which is the confidence interval of a normally distributed value having zero mean and variance $\frac{1}{T}$.

The question is where the $\frac{1}{T}$ is coming from. Why is the variance of the autocorrelation $r_h$ of a white noise process equal to $\frac{1}{T}$?

Hill, 2011:p.349; Chatfield, 2009:p.56; Kendall, 1983 answer the question properly.

Hill, R.C., Griffiths, W.E., Lim, G.C. (2011). Principles of Econometrics, 4.ed., Wiley, USA. https://www.wiley.com/en-us/Principles+of+Econometrics%2C+4th+Edition-p-9780470626733

Chatfield, C. (2009). The Analysis of Time Series: An Introduction, 6.ed., London, Chapman and Hall/CRC. https://www.crcpress.com/The-Analysis-of-Time-Series-An-Introduction-Sixth-Edition/Chatfield/p/book/9781584883173

KENDALL Stuart Ord 1983 The Advanced Theory of Statistics Vol3 4E.