Solving a specific difference equation

Question

Dynamics

I have the following recursive discrete dynamics for $t=1,2,\dots,d$ where we can assume $d>100$ or even $d\to\infty$ if needed.

\begin{align} x_{1} =1,\quad \quad x_{2} = \frac{1+\sqrt{1-4\frac{1}{d}}}{2},\quad \quad \forall t=3,\dots,d:~~ x_{t} =\frac{x_{t-1}+\sqrt{x_{t-1}^{2}-4\beta_{t}}}{2} \end{align} where $ \beta_{t}=\frac{1}{d}\left(\left(t-1\right)2^{-\frac{2t-4}{d}}-\left(t-2\right)2^{-\frac{2t-5}{d}}\right) $.

Overall, $\beta_t$ are rather small (since $d\ge 100$) and strictly positive. Then, at each iteration the decrease $x_{t-1} - x_t$ is also small. The iterates behave as plotted below:

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The problem

1. Ideally, we would hope to find a closed-form for $x_t$.
2. Alternatively, it is also useful for me to prove that $x_{t} - 2x_{t-1}+x_{t-2}>0$, i.e., that $\left(x_{t-1}-x_{t}\right)_{t}$ is monotonically decreasing.
   This seems to be the case numerically, but we couldn't prove it. We tried to solve an ODE with a continuous approximation, but that didn't work out.

Any help would be greatly appreciated!

Answer 1

In the limit $d\to\infty$ the op's difference equation can be transformed into an ordinary differential equation, which can be solved exactly.

We define $\tau=t/d$ and expand $\beta(\tau) = \beta_{\tau d}$ around $d=\infty$, \begin{align}\tag{1}\label{eq:1} \beta(\tau) = \frac{4^{-\tau}(1-\tau \ln2)}{d} + \mathcal{O}(d^{-2})\,. \end{align} Next, we rewrite the $x_t$-recursion in terms of $\tau$, too, and get (with $\delta\tau=1/d$) \begin{align}\tag{2a}\label{eq:2a} x(\tau) &=\frac{x(\tau{-}\delta\tau)+\sqrt{x^2(\tau{-}\delta\tau)-4\beta(\tau)}}{2}\\ \tag{2b}\label{eq:2b} &=\frac{x(\tau{-}\delta\tau)+\sqrt{x^2(\tau{-}\delta\tau)-4^{1-\tau}(1-\tau \ln2)\,\delta\tau+\mathcal{O}(\delta\tau^2)}}{2}\,, \end{align} such that \begin{align} \tag{3a}\label{eq:3a} x'(\tau)&=\lim_{\delta\tau\to 0}\frac{x(\tau)-x(\tau{-}\delta\tau)}{\delta\tau}\\ \tag{3b}\label{eq:3b} &=-\frac{4^{-\tau}(1-\tau \ln2)}{x(\tau)}\,. \end{align} The solution of this ODE with initial condition $x(0)=1$ reads \begin{align} \tag{4}\label{eq:4} x(\tau)=\sqrt{1-\frac{1}{\ln4}+4^{-\tau}\left(\frac{1}{\ln4}-\tau\right)}\,. \end{align} The convexity for $0\leq\tau\leq1$ follows.