By simulation we create a vector $Y = (y_1,y_2,...,y_n)$, where each $y_i \in R$ is independently drawn from a given non-degenerate distribution.

Next we create by simulation a vector $\xi = (\xi_1,\xi_2,...,\xi_n)$ where each $\xi_i$ are independent realizations of a random variable which takes only finite number of values $[\alpha_1,\alpha_2,...\alpha_k]$ with probabilities $p_1,p_2,...,p_k$ respectively. $\alpha_i$ are given.

Suppose that we have got function $f: R \to R$

We make a regression of$\begin{bmatrix} f(y_1+\xi_1) \\ f(y_2+\xi_2) \\ ... \\ f(y_n+\xi_n) \end{bmatrix}$ on $\begin{bmatrix} f(y_1+\alpha_1) & f(y_1+\alpha_2) & ...& f(y_1+\alpha_k) \\ f(y_2+\alpha_1) & f(y_2+\alpha_2) & ... & f(y_2+\alpha_k)\\ ... & ... & ... & ... \\ f(y_n+\alpha_1) & f(y_n+\alpha_2) &... & f(y_n+\alpha_k) \end{bmatrix}$

By regression I mean that we are optimizing $\beta_i$ to minimize:


Intuitively I think that as $n \to \infty$ least squares procedure should give us the following equation:

$f(Y + \xi) = p_1*f(Y+\alpha_1) + p_2*f(Y+\alpha_2) + ... +p_k*f(Y+\alpha_k)$

where $f(Y + \xi)$ and $f(Y+\alpha_i)$ are just representations of vector columns above.

So my conjecture is that as $n \to \infty, \beta_i \to p_i$.

My question is what conditions should be imposed on function $f$ to get the equation above? Maybe $f$ should be smooth enough etc.Is my intuition correct that normally we should get such a equation? Maybe we need to impose some conditions on the distribution of $y_i$ also like $y_i$ is bounded etc.


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