# Is sum of products of random variables is Martingale

Problem
We are given two sequences of i.i.d random variables $$A_1, A_2,\dots$$ and $$B_1, B_2, \dots$$ with known expected values $$\mu_A$$, $$\mu_B$$ and variances $$\sigma^2_A$$, $$\sigma^2_B$$. Define new variable $$C_t$$ that at time $$t$$ takes a value $$C_t = \sum_{i=1}^t A_i\cdot B_i$$ Is $$C_t$$ a Markov process and when it is a martinagle?

My ideas
I guess that $$C_t$$ is indeed a Markov process because the probability of $$C_t$$ taking some value depends only the value of the previous sum, $$\sum_{i=1}^{t-1} A_i\cdot B_i$$ which is $$C_{t-1}$$. However, the proof is more intuitive than formal.

As for the second part, here I'm less certain. If I'm not mistaken, $$E(C_t | C_{t-1}) = C_{t-1} + \mu_A\cdot \mu_B$$. So, $$C_t$$ is martingale if $$E(C_t | C_{t-1}) = C_{t-1}$$ or $$\mu_A\cdot \mu_B = 0$$?

• Your guesses are right. It is always Markov and it is a martingale iff $EA_iEB_i=0$. Oct 11, 2022 at 11:23
• Is there a more formal way to show that it's Markov? Oct 11, 2022 at 12:25
• All you need is the fact that $A_{t+1}B_{t+1}$ is independent of $C_t$ Oct 11, 2022 at 12:30