1. Find a counter example shows that n, does not imply
Let denote a sequence of i.i.d. exp(1) random variables. For and any n, such that , where . But the event is equivalent to . From exponential we see that so is just the n-th order statistic of the random variables . The CDF of this order statistic is , so . We cannot find any finite such that for . So a counter example is found.
2. Proof that if converge to random variable , then converge to 0 in probability.
By the statement in Portmanteau Lemma in the textbook, for any fixed M satisfies , exceeds arbitrarily little for large n. Then , such that when we have . By the triangular inequality, we have . . As n goes to infinity, we have goes to 0. So converge to zero in probability.
3. Problem 3.2
Define a function from a subset of to .
Then is a continuous function. Let be the bivariate sample of sample size n. Without loss of generality, suppose , and with being the correlation coefficient. Let , ,, , . Then
Here we have , , and . Let and . Then
Let , , . A is a certain constant matrix. It is easy to see that and are linear functions of , , and , which are all finite. Thus
What's more, we will take partial derivative with respect to each variable in the function and evaluated at , which are
So according to Delta method,