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Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this process. Is it always true that a distribution of observed sequence itself converges to a normal distribution? I understand that if mean and variance are arbitrary functions then it is easy to construct counterexamples, but what about the case of "nice" functions?

tosik
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  • The _realization_ is a sequence of real numbers and as such, _if_ the sequence converges at all, it converges to a real number and not to a distribution, whether normal or not. – Dilip Sarwate Feb 25 '19 at 16:42
  • @DilipSarwate I've edited the question. What I mean in particular is for example if I simulate a Vasicek process and plot a histogram of one realization, then it looks normal. My question is whether it is always the case and, if yes, what is the underlying theory for it? – tosik Feb 25 '19 at 17:02
  • No, the histogram of values is not necessarily one that will "always look normal" even for the case when $\mu(t)$ and $\sigma^2(t)$ are "nice" functions. Of course, _constants_ $\mu_0 = \mu(t) \forall t$ and $\sigma_0^2 = \sigma^2(t) \forall t$ are "particularly nice" functions and when you have a histogram of values drawn from a $N(\mu_0, \sigma_0^2)$ distribution, by golly, that histogram is gonna "look normal." – Dilip Sarwate Feb 25 '19 at 22:14
  • @DilipSarwate Even if nice $\mu(t)$ and $\sigma(t)$ converge to some values $\mu$ and $\sigma$? – tosik Feb 26 '19 at 09:02
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    @tosik: what do you mean when you say you plotted a realization ? do you mean that you plotted the trajectory of one simuiation ? It's not clear to me what the horizontal axis is when you plot the histogram ? – mlofton Feb 26 '19 at 22:54
  • @mlofton I simulate one trajectroy, then plot histogram of these simulated values. – tosik Feb 27 '19 at 09:13
  • I'm honestly not sure. Take brownian motion for example. if you plot one trakectory of brownian motion and constructed a histogram of the intermittent locations along that path, that won't look normal. But, if you plotted the increments ( distance between current location and previous location ), they would. So, I'm not sure if that's a counter-example but it might be because browniam motion is "in some informal sense" N(0,t). Dilipe: Is this a counter-example ?.. – mlofton Feb 27 '19 at 23:49

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