Say let $Y_1, Y_2,...$ be a family of i.i.d random variables with a common $\mu$ and common variance $\sigma^2$. Let $N_t$ be a Poisson process with rate $\lambda >0 $. Assume that $N_t$ is independent of the $Y_i 's$. Consider the stochastic process given by $$X_t = \sum_{i=1}^{N_t}{Y_i}$$.

So if say $Y_i's$ are discrete random variables with common p.m.f $P\left\{Y_i=\alpha_j\right\} = p_j , j=0, 1, ...$ , $\sum_{j=0}^{\infty}{p_j} = 1$ and the $\alpha_i 's $ are fixed real numbers. Assue that each time an event of $N_t$ occurs it is classified as an event of type j if the number $\alpha_j$ is added to $X_t$, i.e., the ith event is classified as of type j if $Y_i = \alpha_j$. Let $N_j(t)$ be the number of type j events occuring by time t, j = 0, 1, 2, ...

My question is:

How can $X_t$ defined above be written as $\displaystyle\sum_{j=0}^{\infty} \alpha_jN_j(t)$ ? i.e. how is it that $X_t = \displaystyle\sum_{j=0}^{\infty} \alpha_jN_j(t)$?

Could someone give some explanations or comments?

  • $\begingroup$ It might be useful to explain how do you know that $X_t=\sum_{j=0}^\infty \alpha_j N_j(t)$. $\endgroup$ – Juho Kokkala Nov 21 '18 at 19:21
  • $\begingroup$ Yes how do you know what $X_t$ is actually equal to that is my question. Thanks $\endgroup$ – john_w Nov 21 '18 at 21:46
  • $\begingroup$ I meant: how did you come up with the question? To be able to write the question, the conjecture $X_t=\sum_{j=0}^\infty \alpha_j\,N_j(t)$ must have entered your mind from somewhere. If you read this from some paper, please give a reference. If this is a conjecture based on some intuitive reasoning or e.g. simulations, please explain. Or, to rephrase my first comment: it might be useful to explain why you think that $X_t=\sum_{j=0}^\infty \alpha_j N_j(t)$ might be true. $\endgroup$ – Juho Kokkala Nov 22 '18 at 6:17
  • $\begingroup$ Hello, I actually not smart enough to come up with that. But it is just a question that I read from a book. Basically it says the $X_t$ can be re-written as the very last summation format. $\endgroup$ – john_w Nov 23 '18 at 17:20
  • $\begingroup$ "I read from a book" is an answer to the question I tried to ask -- please add the reference (preferably mentioning page number, exercise number if this is an exercise question etc.). Thought if this is an exercise question in a textbook, guidelines at stats.stackexchange.com/tags/self-study/info apply $\endgroup$ – Juho Kokkala Nov 23 '18 at 19:32

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