For a given input signal, for each discretized sample, $dt=1$: $$ x(t) \sim N(0,1)$$
and a given LTI low order filter with transfer function: $$ H_{fil}(f)$$
which would be is the a priori distribution of the filtered signal? $$ x_{fil}(t) \sim ? $$
this is, not the conditional distribution of the 1 step predictor: $$ x_{fil}(t)|x(t-1) \sim ? $$
but the distribution of the $n=t/dt$ steps simulation: $$ x_{fil}(t)|x(0) \sim ? $$
For example, the special case of the integration: $$ H_{int}(f)=\frac{1}{2\pi f}$$
has a distribution of the sum of gaussian distribution: $$ x_{int}(t) \sim N(0,\sqrt{\frac{t}{dt}}) $$
this is a normal distribution, with a variance which increases linearly with the number of samples.
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