# Convergence of regression coefficients to probability density

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:

$$\sum_{i=1}^n(f(Y+\xi)-\sum_{j=1}^k\beta_jf(Y+\alpha_j))^2$$

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.