1.

a)

See Question 8 of Chapter 3.

b)

First, note that

where and . By the WLLN, and converge, in probability, to and . By Theorem 5.5 (d), and converge, in probability, to the same quantities. Lastly, by Theorem 5.5 (a), converges, in probability, to .

2.

Suppose converges to in quadratic mean. By Jensen’s inequality,

Therefore, .

Next, note that

Taking limits of both sides reveals . The converse, we can apply the limits and directly to the equation above.

3.

First, note that

Taking the limit,

4.

Let . For sufficiently large,

and hence converges in probability. However,

and hence does not converge in quadratic mean.

5.

It is sufficient to prove the second claim since convergence in quadratic mean implies convergence in probability. First, note that

Taking expectations, and using the fact that and ,

6.

Letting denote the CDF of a standard normal distribution, by the CLT,

7.

Let be a function and be a constant. Then,

It follows that converges to zero in probability. Take for Part (a) and for (b).

8.

Letting denote the CDF of a standard normal distribution, by the CLT,

9.

Let . Then,

Therefore, converges in probability (and hence in distribution) to . On the other hand,

10.

Since whenever , it follows that

Therefore,

11.

First, note that is almost surely zero. Let and be a standard normal random variable. Then,

Therefore, converges in probability (and hence in distribution) to zero.

12.

Let be the CDF of an integer valued random variable . Let be an integer. It follows that for all . We use this observation multiple times below.

To prove the forward direction, suppose . By definition, at all points of continuity of . Therefore,

To prove the reverse direction, suppose for all integers . Let be an integer and note that

and hence as desired.

13.

First, note that

If , it follows that . Otherwise,

Therefore, and hence converges in distribution to an random variable.

14.

By the CLT

Let so that . By the delta method,

15.

Define by . Then, . Define for brevity. By the multivariate delta method,

16.

Let be IID with . Trivially, and . However, while and hence does not converge in distribution to .