This is a basic question I came across when I started learning random processes. Suppose I have a random variable $f(t) = G(t) + n(t)$ where $G(t) = A\exp(-t^2/T^2)$ (i.e. a deterministic function having Gaussian shape) and $n(t)$ a stationary random variable with uniform probability distribution between $-B$ and $B$ (hence it has zero mean).
Now $f(t)$ is clearly a nonstationary random variable because its ensemble average is time dependent and is equal to $G(t)$. Next I want to connect it with the concept of ergodicity which I am also new to. According to the book "Random Data - Analysis and Measurement Procedures" by Bendat and Piersol, they say that if the time average and autocorrelation function of the k-th sample function of a random process do not differ when computed over different sample function (different k), then the random process is ergodic. However in my example the time average of $f(t)$ for a k-th sample function $$ \mu_f(k) = \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T f_k(t) dt = \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T G(t) dt $$ seems to be independent of k. If this were true then this random variable is ergodic, but it is nonstationary. Where am I mistaken? Second thing, about the definition of time average, should the lower limit for the integral indeed be $0$ or $-T$?