How is the averaging of this stochastic process determined?

Question

I am reading a physics paper dealing with a random fluctuation $R(t)$ satisfying:

1. $\langle R(t)\rangle = 0$.
2. $\langle R(t) R(t')\rangle = \delta(t - t')$ where $\delta$ is the Dirac-delta function

The paper says:

$R(t)$ is not a well-behaved function. Instead of its instantaneous value, which is not well-defined, one needs to take its average over the interval $\delta t$, which depends on $\delta t$ and is $r/\sqrt{\delta t}$, where $r$ is a random number with $\langle r \rangle = 0$ and $\langle r^2\rangle = 1$.

My Question is: How was this average determined?

Thank you for your response.

Answer 1

In this paper, $R(t)$ is a gaussian white noise which is formally the derivative of a Wiener process. When integrated over time $\delta t$, it becomes a normally distributed random variable with mean zero and standard deviation $\sqrt{\delta t}$. So its average, (its integral over $\delta t$) would be $1/\sqrt{\delta t}$ times a normalized random variable that is called $r$ here.