# consistency of an estimator not based on total sample size

How do I show the consistency of an estimator of a parameter, say $$\mu$$, that is not based on the sample size $$n$$ but a function of $$n_{i}$$'s where $$\sum_{i=1}^{K}n_{i}=n$$ ? Consider for example the estimator $$\hat{\mu}=\frac{1}{K}\sum_{i=1}^{K}\frac{Y_{i}}{n_{i}}$$ where $$Y_{i}$$'s are independent r.v.'s with mean $$n_{i}\mu$$ and variance $$\sigma^{2}n_{i}$$.

• Do $n_i\to\infty$? Or $K\to\infty$? What's the relationship between $n_i$'s? – Julius Vainora Dec 12 '18 at 0:39
• $K$ is finite. We know $\sum_{i=1}^{K}n_{i}=n$ where $n_{i}>0$. How do I define a sequence of estimators based on $\hat{\mu}$ and $n$ to check its consistency ? The definition of consistency involves $n$. – Sayan Dec 12 '18 at 0:43