A random sample size is observed together with the data and may carry information about the parameter. When it is external or asymptotically deterministic, fixed-sample LAN survives under a time change. When it is data dependent, martingale likelihood theory and stable convergence are needed.
The random-horizon experiment
Definition. For an integer-valued \(N_n\), the observed data are \[ (N_n,X_1,\ldots,X_{N_n}). \] If \(N_n\) is a stopping time, this is the stopped \(\sigma\)-field \(\mathcal F_{N_n}\). If \(N_n\) is external and independent, it is an ancillary randomization of information time.
Anscombe's random-index theorem
Theorem. Let \(S_k=\sum_{i=1}^kY_i\), where \(Y_i\) are i.i.d., centered, with variance \(\sigma^2\). If \[ \frac{N_n}{n}\to_p\tau>0 \] and the partial-sum process satisfies the usual stochastic equicontinuity in time, then \[ \frac{S_{N_n}}{\sqrt n} \rightsquigarrow N(0,\tau\sigma^2). \]
Proof. Donsker's theorem gives \[ t\mapsto\frac{S_{\lfloor nt\rfloor}}{\sqrt n} \rightsquigarrow\sigma W(t) \] in \(D[0,\infty)\). Jointly, \(N_n/n\to_p\tau\). Brownian paths are continuous, so evaluation at the random time is a continuous mapping at the limit: \[ \frac{S_{N_n}}{\sqrt n} \rightsquigarrow\sigma W(\tau) \sim N(0,\tau\sigma^2). \] \(\square\)
Random-size LAN
Theorem. Suppose an i.i.d. model is DQM at \(\theta_0\) with information \(I\), and \(N_n/n\to_p\tau>0\) under \(\theta_0\). Under conditions allowing the stopped score and likelihood remainder to obey Anscombe-type convergence, \[ \log\frac{\mathrm{d}P_{\theta_0+h/\sqrt n}^{\,N_n}} {\mathrm{d}P_{\theta_0}^{\,N_n}} = h^\top\Delta_{N_n,n} -\frac12\tau h^\top Ih+o_p(1), \] where \[ \Delta_{N_n,n} = \frac1{\sqrt n}\sum_{i=1}^{N_n}\dot\ell_{\theta_0}(X_i) \rightsquigarrow N(0,\tau I). \] Thus information at the \(\sqrt n\) scale is \(\tau I\).
If one instead normalizes by \(\sqrt{N_n}\), the information returns to \(I\), provided \(N_n\to\infty\).
Random information fractions
Proposition. If \[ N_n/n\rightsquigarrow T>0 \] stably, with nondegenerate random \(T\), then the central sequence can have mixed normal limit \[ \Delta_{N_n,n}\rightsquig T^{1/2}I^{1/2}Z, \] and the local experiment is LAMN with random information \(TI\).
Stable convergence is needed because the random information \(T\) remains part of the limit experiment and may be related to ancillary or design variables.
Data-dependent stopping
If \(N_n\) depends on the data, the likelihood ratio is the stopped likelihood martingale, not merely a fixed-\(N_n\) density with \(N_n\) treated as external. Martingale central limit conditions require predictable quadratic variation: \[ \langle M\rangle_{N_n}/n\to_p\tau I \] and a conditional Lindeberg condition. Optional stopping must preserve likelihood-ratio mean to retain contiguity.
Informative sample sizes
Example. Suppose sampling stops early when observations are large. Then \(N_n\) itself is associated with the parameter, and conditioning on \(N_n\) can discard information or alter the sampling law. Treating the realized size as if it had been fixed in advance may give correct likelihood ratios under a stopping-rule principle but incorrect unconditional frequentist calibration.
Group-sequential designs
At predetermined information fractions \[ 0<t_1<\cdots<t_K=1, \] the standardized score vector converges to Gaussian Brownian increments with covariance \[ \operatorname{Cov}\{\Delta(t_j),\Delta(t_k)\} =(t_j\wedge t_k)I. \] Stopping boundaries are calibrated against this joint law. Repeatedly applying a fixed-sample critical value would inflate type-I error because the looks are correlated opportunities to reject.
Cost and information
A sequential rule can reduce expected sample size under favorable parameter values while preserving error probabilities. It does not create Fisher information for free: expected information, sampling cost, and terminal loss must all appear in the decision criterion. Fixed-sample efficiency and sequential optimality answer different questions.