# probabilistic distribution of given data

let us consider following model

$$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$

we have three parameter fixed,but unknown and also $z(t)$ is simple white noise, before i will ask my question let us consider following situation,because parameters are fixed, that means that sinusoidal components represents deterministic model right?even more if we know these parameters,then at some time $t=t_0$, we can consider each model as scalar so we have sum of scalars + random variable, because white noise are iid, does it means that sum also will be iid? let us consider following case,we have $p$ deterministic component,let us construct following equation

$\alpha_{11}+\alpha_{12}+\cdots+\alpha_{1p}+z(1)$

$\alpha_{21}+\alpha_{22}+\cdots+\alpha_{2p}+z(2)$

$\vdots$

$\alpha_{n1}+\alpha_{n2}+\cdots+\alpha_{np}+z(n)$

we can consider it as

$$y_1,y_2,\ldots,y_n$$

now how can i estimate what is a probability distribution form of $y_i$? let us suppose that original sinusoidal components are just harmonically related to each other,does it help us to determine distribution form of outcomes? what if they are not harmonically and ratio of two frequency can be irrational? i know there exist for example

• Markov process

• Lévy process

• A renewal process

and so on. how can i find relevant model for this process? thanks in advance