Walk with randomised boosts

Question

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 $t$?

Remark #1. 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.

Remark #2. I originally requested an answer regarding the density function of the continuous version $x(t)$ as a fall-back. I have accepted an answer on that question for now; and that answer implicitly provides an answer for the discrete-time version in which the steps are normally distributed instead of being of fixed size. However, I will preferentially accept and reward any answer on the discrete-time version of the problem.

Answer 1

The position process $x_t$ satisfies $$ x_t = x_0 + t v_0 + \int_0^t W_s ds \;. $$ Because $\int_0^t W_s ds \sim \mathcal{N}(0,\frac{1}{3} t^3)$, a simple change of variables shows that $$ x_t \sim \mathcal{N}( x_0 + t v_0, \frac{t^3}{3}) \;. $$

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The following relates $x_t$ to a discrete-time weak approximation $X_N$, which is obtained by weakly approximating the Brownian increments in the velocity process $v_t$ by Rademacher random variables.

Claim. Given $t>0$ and $N \in \mathbb{N}$, set $$ \tag{$\star$} X_n = X_{n-1} + h V_n \;, \quad V_n = V_{n-1} + \sqrt{h} A_n \;, \quad 1 \le n \le N $$ with initial conditions $X_0 = x_0$, $V_0 = v_0$, and where $\{ A_i \}_{i=1}^N$ are independent Rademacher random variables and $h=\frac{t}{N}$. For any $t>0$, $$ X_{N} \overset{d}{\to} x_t \quad \text{as $N \to \infty$} \;. $$

Proof. Unraveling the recurrence relation ($\star$) yields, $$ X_N = x_0 + t v_0 + \frac{t^{3/2}}{N^{1/2}} S_N $$ where $S_N = \sum_{i=1}^N A_i \frac{(N-i+1)}{N}$. Note that $S_N$ is a sum of independent and uniformly bounded random variables each with mean zero. Moreover, the variance $s_N^2$ of $S_N$ goes to $\infty$ as $N \to \infty$ since $$ s_N^2 = \operatorname{Var}(S_N) = \frac{1+3 N + 2 N^2}{6 N} \;. $$ By the Lyapunov Central Limit Theorem, $$ \frac{S_N}{s_n} \overset{d}{\to} \mathcal{N}(0,1) \;, $$ from which it follows that, $$ X_N \overset{d}{\to} \mathcal{N}(x_0 + t v_0, \frac{t^3}{3} ) \;. $$