Random walk and probability of returning at the starting point

Question

We are given a d -dimensional random walk as a stochastic process $(R_N)_{N\in\mathbb{N_0}}$ with discrete time and values in $\mathbb{R^d}$. We define the displacement $\Delta R_N = R_N-R_0 = \sum_{i=1}^N a_i$ as the result of independent, identically distributed random increments (steps) $a_i \in \mathbb{R^d}$ with probability density $p_a$ which is supposed to be symmetric, $p_a(x) = p_a(-x).$ The distribution of steps with characteristic function $\varphi_a(k)$ shall be symmetric in $k$ and possess a finite second moment. I need to provide an expression for the probability to be at the starting point after exactly $N$ steps in terms of $\varphi_a(k).$ I do not understand what is meant by the assertion that the distribution of steps with characteristic function $\varphi_a(k)$ shall be symmetric in $k$, if we already say at a previous sentence that the distribution is symmetric. I also do not know how to proceed in solving this problem. I will appreciate any help or a solution proposal. Thanks.

Answer 1

Hint

Let $b = \sum a_i$. Check that $b$ is a ($d-$dimensional) random variable that correspons to the total displacement.

Check that you are interested in $P(b=0)$

Try to write the probability function $p_b$ in terms of $p_a$, and the same for the correspoding characteristic functions ($\phi_b$ in terms of $\phi_a$). You'll find the the latter is easier. Write $P(b=0)$ in terms of $\phi_b$.