I am reading "Introduction to Stochastic Processes " by Lawler, and I'm having trouble understanding the proof of the reflection principle for a discrete Martingale (Section 5.6).
Reflection Principle. Let $X_1, X_2,\dots$ be independent random variables whose distribution is symmetric about the origin. Let $M_0=0$ and $M_n=\sum_{i=1}^nX_i$. Let $\bar{M}_n=\max \{ M_0,\dots,M_n \}$Then, for every $a>0$, $P(\bar{M}_n\ge a)\le 2 P(M_n \ge a)$.
The proof uses the following inequality, which I'm having trouble with. For $j\in\{1,\dots,n\}$
$$
P(M_n – M_j \ge 0)\ge 1/2.
$$
Now, if $n=j+1$, then this is obvious, as $X_{j+1}$ is symmetric about the origin, so $P(M_n – M_j \ge 0) = P(X_{j+1}\ge 0)= 1/2$. Furthermore, if $\{X_i\}_i$ are iid and each follows a normal distribution, then $M_n – M_j$ also follows a normal distribution with mean zero, so $P(M_n – M_j \ge 0)= 1/2$. But, given the more general condition given in the statement, I'm not sure how to prove this.
Best Answer
If $(X_n)_{n \in \mathbb{N}}$ be independent random variables with symmetric distribution i.e. $X_n\sim -X_n$. Let $M_n=X_1+...+X_n$. Then $M_n-M_k=\sum_{k<\ell \leq n}X_\ell$ has symmetric distribution. To see this, we note that a rv has symmetric distribution iff its characteristic function is purely real. Now, by independence $E[e^{i\xi (M_n-M_k)}]=\prod_{k<\ell \leq n}E[e^{i\xi X_\ell}]$. So the characteristic function of $M_n-M_k$ is purely real, thus $M_n-M_k$ has symmetric distribution.