# calculate Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$

let $X_1,X_2,\ldots,X_n$ are random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and $Y_1,Y_2,\ldots,Y_n$ are random sample of geometric distribution with parameter of $\theta_1+\theta_2$.If the two samples are independent of each other, how can i calculate Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$

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By the law of large numbers, the ratios converge almost surely to $E(X_1)/E(Y_1)$ hence they converge in distribution to the same value. This uses that $(X_n)$ is i.i.d. and that $(Y_n)$ is i.i.d. but not the independence of $(X_n)$ and $(Y_n)$.