Let X1,X2,… be a Bernoulli process. We will define some new sequences of random variables and inquire whether they form a Bernoulli process.

Question

1. Let Yn = X2n. Is the sequence Yn a Bernoulli process?

2. Let Un = Xn+1. Is the sequence Un a Bernoulli process?

3. Let Vn = Xn+Xn+1. Is the sequence Vn a Bernoulli process?

4. Let Wn = (−1)nXn. Is the sequence Wn a Bernoulli process?

Answer 1

1. Yes
2. Yes
3. No
4. No