2
$\begingroup$

I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas?

Let $X_1, X_2, \cdots$ be an infinite sequence of iid sequence of integrable random variables and let $N$ be a positive, integer-valued integrable random variable which is independent from the sequence. Define $S_n = \sum_{k=1}^{n} X_k$ and assume that $S_0 = 0$.

(a) Show that $E[S_N] = E[N]E[X_1]$.

(b) Show that the characteristic function of $S_N$ is given by $E[\phi_{X_{1}}(t)^N]$, where $\phi_{X_{1}}$ is the characteristic function of $X_1$.

  • 0
    Incidentally, (a) is a special case of Wald's equation (http://en.wikipedia.org/wiki/Wald_equation).2010-12-07

1 Answers 1

4

For part (a) use $E[S_N]= \sum\nolimits_n {E[S_N |N = n]P(N = n)}$. This leads straightforwardly to the result. For part (b) use $E[e^{tS_N } ] = \sum\nolimits_n {E[e^{tS_N } |N = n]P(N = n)} $. Again, this leads straightforwardly to the result.

  • 2
    Note that the key in both parts is "condition on $N$".2010-12-07
  • 0
    Yes, it uses conditional expectation and independence. Its not too difficult after all.2010-12-08