Gaussian Markov-process limit for the birth Mallows jump count

Complete functional limit theorem. Let $J_n(t)$ be the jump count of Corsini's birth Mallows process on $S_n$. For every $T<1$,

$$\left(\frac{(1-t)J_n(t)-nt}{\sqrt n}\right)_{0\le t\le T} \Longrightarrow (W_t)_{0\le t\le T} \qquad\text{in }D([0,T]),$$

where $W$ is standard Brownian motion. Thus the conjectured Gaussian Markov limit exists and is explicitly identified, with covariance $\mathbb E[W_sW_t]=\min(s,t)$. The convergence also holds locally on $D([0,1))$.

The source constructs the process using independent inversion-coordinate birth processes $B_j(t)$, $1\le j\le n$, with

$$J_n(t)=\sum_{j=1}^n B_j(t),\qquad \Pr(B_j(t)=r)=\frac{(1-t)t^r}{1-t^j} \quad(0\le r<j).$$

The finite birth rate simplifies exactly to

$$q_j(t,r) =\frac{r+1}{1-t} -\frac{j\,t^{j-r-1}(1-t^{r+1})} {(1-t)(1-t^j)}.$$

Indeed, because jumps have unit size, the probability current across state $r$ is

$$q_j(t,r)\Pr(B_j(t)=r) =-\frac{d}{dt} \frac{1+t+\cdots+t^r}{1+t+\cdots+t^{j-1}},$$

which gives the displayed formula directly. In particular,

$$0\le q_j(t,r)\le\frac{r+1}{1-t}.$$

Let $Y_1,Y_2,\ldots$ be independent Yule birth processes started at zero, with jump rate

$$q_\infty(t,r)=\frac{r+1}{1-t}.$$

Their laws are $\Pr(Y_j(t)=r)=(1-t)t^r$. Independently for each $j$, couple $B_j$ to $Y_j$ by accepting a Yule jump, when the current states are $b\le y$, with probability

$$\frac{q_j(t,b)}{(y+1)/(1-t)}.$$

The accepted intensity is exactly $q_j(t,b)$. Therefore $D_j(t)=Y_j(t)-B_j(t)$ is nonnegative and nondecreasing. The exact geometric means give

$$\mathbb ED_j(T) =\mathbb EY_j(T)-\mathbb EB_j(T) =\frac{jT^j}{1-T^j}.$$

Consequently,

$$\mathbb E R_T :=\mathbb E\sum_{j\ge1}D_j(T) =\sum_{j\ge1}\frac{jT^j}{1-T^j} \le\frac{T}{(1-T)^3}<\infty.$$

Hence $R_T<\infty$ almost surely, and for every $n$,

$$\sup_{0\le t\le T} \left|J_n(t)-\sum_{j=1}^nY_j(t)\right| \le R_T.$$

In particular, the normalized path discrepancy is $O(n^{-1/2})$ almost surely.

Now set

$$X_j(t)=(1-t)Y_j(t)-t,\qquad Z_n(t)=\frac1{\sqrt n}\sum_{j=1}^nX_j(t).$$

The birth intensity makes $X_j$ a centered square-integrable martingale:

$$dX_j(t)=(1-t)\,dY_j(t)-(Y_j(t)+1)\,dt.$$

Its predictable bracket is

$$\langle X_j\rangle_t =\int_0^t(1-s)(Y_j(s)+1)\,ds.$$

Since

$$\mathbb E[(1-s)(Y_j(s)+1)]=1,\qquad \operatorname{Var}((1-s)(Y_j(s)+1))=s,$$

independence gives the quantitative uniform bracket bound

$$\mathbb E\sup_{t\le T} \left|\langle Z_n\rangle_t-t\right| \le\frac1{\sqrt n}\int_0^T\sqrt s\,ds =\frac{2T^{3/2}}{3\sqrt n}.$$

Furthermore, $\sup_{t\le T}|\Delta Z_n(t)|\le n^{-1/2}$. The martingale functional central limit theorem therefore yields $Z_n\Rightarrow W$ in $D([0,T])$, with $W$ standard Brownian motion. Finally,

$$\sup_{t\le T} \left| \frac{(1-t)J_n(t)-nt}{\sqrt n}-Z_n(t) \right| \le\frac{R_T}{\sqrt n}\longrightarrow0 \quad\text{almost surely}.$$

Slutsky's theorem proves the stated functional convergence for every $T<1$.

Prior-work distinction. The Yule thinning itself appears in Adamczak–Kotowski (2025), which proves different global and local limits. The argument above uses its summable error to establish the missing Brownian functional fluctuation theorem. Dubach (2026) analyzes a different coupling and explicitly leaves extensions to regular Mallows processes and Skorohod convergence open.

Original statement: Corsini's arXiv version, Conjecture 6, §4.3; the revised author manuscript renumbers the same statement as Conjecture 5.