# Is the sequence of uniformly distributed random variables martingale?

Let $$(Y_n, n\ge 1)$$ is the sequence of independent, uniformly distributed on the interval $$[-1, 1]$$ random variables. $$S_n = Y_1 + Y_2 + ... + Y_n,$$ $$n\ge 1$$ and $$S_0 = 0.$$

If the sequences:

a) $$X_n = \sum_{k=1}^n S_{k-1}^2 Y_k$$, $$X_0 = 0$$,

b) $$X_n = S_n^2 - \frac{n}{3}$$, $$X_0 = 0$$

are martingales?

My solution I started from b):

$$\mathbb E[X_{n+1} | S_0, ..., S_n]=\mathbb E[(S_{n}+Y_{n+1})^2 - \frac{n-1}{3} | S_0, ..., S_n]=\mathbb E[S_{n}^2+2S_n Y_{n+1} + Y_{n+1}^2- \frac{n-1}{3} | S_0, ..., S_n]=S_{n}^2 +2S_n \mathbb E[Y_{n+1} | S_0, ..., S_n] + \mathbb E[Y_{n+1}^2| S_0, ..., S_n] - \frac{n-1}{3}=S_n^2 + \frac{1}{3} - \frac{n-1}{3} = X_n.$$

Is it correct? How to solve a)?

You can solve (a) in a similar fashion, but it may be easier to first note that $$X_n$$ is a function of $$S_0, \dots, S_n$$ (write $$Y_k = S_k - S_{k-1}$$). Then to show it is a martingale, it is equivalent to show that $$\mathbb{E}[X_{n+1} - X_n \mid S_0, \dots, S_n] = 0$$, which saves you from having to work with the sum.
Each of these illustrates a common way of creating martingales. (a) is an example of a martingale transform or discrete stochastic integral. For (b), the key is that $$S_n$$ is also a Markov chain, and the function $$f(x,n) = x^2-n/3$$ solves the discrete-time backward heat equation corresponding to the transition function of $$S_n$$.