To find the limit in probability of the sample average πβ―β―β―β―β―π, we can use the Law of Large Numbers. The Law of Large Numbers states that the sample average of a sequence of independent and identically distributed random variables converges in probability to the expected value of the random variable.
In this case, since π1, π2, ..., ππ are i.i.d. Gamma random variables with parameters (πΌ, 1/πΌ), the expected value of each ππ is πΌ/π½ = πΌ. Therefore, the limit in probability of πβ―β―β―β―β―π as π approaches infinity is π = πΌ.
To obtain a consistent estimator πΌΜ of πΌ in terms of πβ―β―β―β―β―π, we can simply use the sample average itself. Therefore, πΌΜ = πβ―β―β―β―β―π.
For the Delta method to apply, the function π needs to be continuously differentiable at a certain value π₯. In this case, we need to find the value of π₯ at which π is continuously differentiable. However, the function π is not provided in the given question, so we cannot determine the specific value of π₯.
Finally, to find the asymptotic variance of πΌΜ, denoted as π(πΌΜ), we would need more information about the estimator πΌΜ and the distribution of the random variables π1, π2, ..., ππ. However, this information is not provided in the question, so we cannot determine π(πΌΜ).