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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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