Boundary crossing problems for sample means. (English) Zbl 0642.60018

Lai, Tze Leung

Ann. Probab. 16, No. 1, 375-396 (1988).

Author’s abstract: Motivated by several classical sequential decision problems, we study herein the following type of boundary crossing problems for certain nonlinear functions of sample means. Let \(X_ 1,X_ 2,..\). be i.i.d. random vectors whose common density belongs to the k-dimensional exponential family \(h_{\theta}(x)=\exp (\theta 'x- \psi (\theta)\}\) with respect to some nondegenerate measure \(\nu\). Let \(\bar X{}_ n=(X_ 1+...+X_ n)/n\), \({\hat \theta}_ n=(\nabla \psi)^{-1}(\bar X_ n)\), and let \(I(\theta,\lambda)=E_{\theta} \log \{h_{\theta}(X_ 1)/h_{\lambda}(X_ 1)\}\) \((=\) Kullback-Leibler information number). Consider stopping times of the form \[ T_ c(\lambda) = \inf \{n: I({\hat \theta}_ n,\lambda)\geq n^{-1} g(cn)\},\quad c>0, \] where g is a positive function such that g(t)\(\sim \alpha \log t^{-1}\) as \(t\to 0\). We obtain asymptotic approximations to the moments \(E_{\theta} T^ r_ c(\lambda)\) as \(c\to 0\) that are uniform in \(\theta\) and \(\lambda\) with \(| \lambda -\theta |^ 2/c\to \infty\). We also study the probability that \(\bar X{}_{T_ c(\lambda)}\) lies in certain cones with vertex \(\nabla \psi (\lambda)\). In particular, in the one-dimensional case with \(\lambda >\theta\), we consider boundary crossing probabilities of the form \( P_{\theta}\{{\hat \theta}_ n\) are: 1) to solve the eigenvalue problems of the Perron-Frobenius operator; 2) to use the Fredholm determinant and 3) to use the renewal equation of words.
We will consider the above problems along the third way, that is, for an irreducible, piecewise linear mapping we will characterize the density of the invariant probability measure and the decay rate of the correlation by zero points of the determinant of a matrix.

MSC:

60F10	Large deviations
62L15	Optimal stopping in statistics
60G40	Stopping times; optimal stopping problems; gambling theory
60G50	Sums of independent random variables; random walks
62L05	Sequential statistical design

Keywords:

sequential decision problems; boundary crossing problems; exponential family; stopping times; eigenvalue problems of the Perron-Frobenius operator; renewal equation; piecewise linear mapping; invariant probability measure