The classical random walk can be described as the evolution of the position $ X_t$ of a walker for integers $ t \geqslant 0$ , where $ X_0 = 0$ and $ X_t = X_{t-1} + V_t$ for $ t \geqslant 1$ , where the “speed” $ V_t$ at each time step is uniformly random $ V_t \in_{\mathrm R} \{-1,+1\}$ and independent at each time step. It is well-known that this process yields a position which obeys a symmetric binomial distribution with mean $ 0$ and variance $ t$ , and that $ t^{-1/2} X_t$ tends to a Gaussian distribution with variance $ 1$ .

I am interested in a variant in which the speed itself increases or decreases by random “boosts”, behaving like a classical random walk, and where the position is governed by the speed as it evolves over time. That is, we have $ $ \begin{aligned} X_0 &= 0 & V_0 &= 0 \ X_t &= X_{t-1} + V_t & V_t &= V_{t-1} + A_t & A_t \in_{\mathrm R} \{-1,+1\}. \end{aligned}$ $ Question. What is the probability distribution of $ X_t$ as a function of $ t$ ?

Observation. This process is in effect a discrete-time variant of a second-order stochastic differential equation of a form $ $ \begin{aligned} \frac{\mathrm dx}{\mathrm dt} &= v, & \frac{\mathrm dv}{\mathrm dt} = \xi_t \end{aligned}$ $ where $ W_t = \int_{0}^t \mathrm d\tau \, \xi_\tau\,$ ; though I do not know enough to be able to distil what probability density function one would expect for $ x(T)$ from the references I found. Failing an answer for the discrete-time case, I will accept an answer for the probability density function of $ x(T)$ .