Finding the limiting probability distribution

Asked Oct 10 '14 at 11:54

Active Oct 10 '14 at 18:47

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I found this problem in Shiryaev's Problems in probability (Problem 3.4.14).

Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = \xi_1 + \dots + \xi_n, \;n \ge 1$, find the limiting probability distribution (as $n \rightarrow \infty $) of the random variables $$\frac1n \sum_{k= 2}^{n}|S_{k - 1}|(\xi_k^2 - 1), \quad n \ge 2.$$

Could anyone give me a hint for solving this? Thank you!

edited Oct 10 '14 at 18:47

asked Oct 10 '14 at 11:54

Ivan Petrarka

Should the lower limit of the sum be $k = 1$ and what is $S_0$? – Ritz Oct 10 '14 at 12:02
I suppose $S_0 = 0$. Fixed the limit – Ivan Petrarka Oct 10 '14 at 12:05
Then let the lower limit be $k = 2$ to avoid confusion. – Ritz Oct 10 '14 at 12:21

Finding the limiting probability distribution

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