# question regarding a stochastic process

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?

• It might be useful to explain how do you know that $X_t=\sum_{j=0}^\infty \alpha_j N_j(t)$. – Juho Kokkala Nov 21 '18 at 19:21
• Yes how do you know what $X_t$ is actually equal to that is my question. Thanks – john_w Nov 21 '18 at 21:46
• 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. – Juho Kokkala Nov 22 '18 at 6:17
• 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. – john_w Nov 23 '18 at 17:20
• "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 – Juho Kokkala Nov 23 '18 at 19:32