8.30 Let Z0,Z1, . . . be independent N(0, 1) random variables. Find a suitable sequence of normalizing constants, an, such that 1

an

n i=1 Zi−1Zi

−→d N(0, 1)

and justify your answer. For the justification part, note that Example 8.14 is often useful in establishing (8.31). Also, note the following facts (and you do need to verify them):

(i) For anyM >0, we have max 1≤i≤n

|Zi−1Zi| ≤ 2



M2 +

n i=0 Z2 i 1(|Zi |>M)



.

(ii) max1≤i≤n(Zi−1Zi )2 ≤



n i=0 Z4 i .