# Convergence of weighted average with fixed variables

I am dealing with administrative data. Suppose I have a finite population of size NN. The finite population estimating equation that defines the parameter θ0\theta_0 can be written as

1NN∑i=1Si(yi,θ) suchthat1NN∑i=1Si(yi,θ0)=0
\frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta) \,\,\,\ such \,\, that \,\,\,\,\, \frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta_0)=0 \tag{1}

Suppose the score function Si(yi,θ)=yi−θS_i(y_i, \theta) = y_i-\theta then the parameter of interest θ0=1N∑Ni=1yi=ˉY\theta_0=\frac{1}{N} \sum_{i=1}^{N}y_i = \bar Y. I want to estimate the parameter ˉY\bar Y considering nonignorable non-response.
Denote the unknown response probability by πi\pi_i and suppose the empirically estimated non-response probability is ˆπi\hat \pi_i. Then the estimating equation for respondent population is

1NN∑i=1δiˆπiSi(yi,θ),
\frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i} S_i(y_i, \theta), \tag{2}

where δ\delta is response indicator and is a random variable, but y′sy’s are now fixed like design based approach in survey sampling.

Taking the expectation of (2) over δ\delta, we have
Eδ[1NN∑i=1δiˆπiSi(yi,θ)|y]=1NN∑i=1Si(yi,θ)Eδ[δiˆπi]
E_\delta[\frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i}S_i(y_i, \theta)|y]= \frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta)E_\delta[\frac{\delta_i}{\hat \pi_i}] \tag{3}

Expanding 1/ˆπ1/\hat \pi around πi\pi_i using Taylor expansion and taking expectation over δ\delta, suppose we have Eδ[δi/ˆπi]=wiE_\delta[\delta_i/\hat \pi_i]=w_i then from (3), we can write

Eδ[1NN∑i=1Si(yi,θ)|y]=1NN∑i=1wiSi(yi,θ)=∑Ni=1wiN∑Ni=1wiSi(yi,θ)∑Ni=1wi
E_\delta[\frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta)|y]=\frac{1}{N} \sum_{i=1}^{N}w_i S_i(y_i, \theta) = \frac{\sum_{i=1}^{N}w_i}{N} \frac{\sum_{i=1}^{N}w_i S_i(y_i, \theta)}{\sum_{i=1}^{N}w_i} \tag{4}

Now wiw_i are positive values/weights and for N \to \inftyN \to \infty, \sum_{i=1}^{N}w_i \to \infty \sum_{i=1}^{N}w_i \to \infty and further it can be assumed that for positive constant C, |w_i|