Limiting Behavior of A Sequence of Density Ratios

Abstract

Let X₁, X₂,".. be a sequence of random variables andР= {P_θ,θ (-)} be a family of distributions of the sequence.For each n,An is the -field generated by X₁,", X_n.If θ₁,θ₂ (-), we define R_n(θ₁,θ₂) as the density ratio of P_θ1,P_θ2onAn.

The main purpose of the paper is to investigate limiting behavior of the sequence R_nwith respect to any P_θ. This has applications in sequential analysis, where it is desired to know whether a sequential probability ratio test terminates with probability one.The same conclusion can be drawn in the case of generalized sequential probability ratio test, under some restriction as to how the stopping bounds vary with n.

If the X_iare independent and identically distributed, then we can write in Rnaswhere the Y_iare independent and identically distributed.We have then thatconverges to ~ or to -~ a.e. according as E_θ{Y_i}>0 or <0. For any θ, say θ₀, for which Eθ₀{Yi}=0 we have lim inf Rn="0" and lim sup R_n=~ a.e. Pθ_0.

A sequence of non-independent nor identically distributed random variables {Xi} may arise in tests of composite hypotheses in the presence of nuisance parameters. An example of the situation is the sequential t-test, by some authors called the WAGR test. In this example we have the same qualitativeresult as if theXi are independent and identically distributed.

The foregoing example suggested the more general problem with the assumption Aand B (see chapters 2 and 3). The result can be described as follows: If θ₁< θ₂then R_nconverges a.e. to 0 if θθ₁and to ~ if 0 θ₂. For θ between θ₁and θ₂, except perhaps for one θ₀, then lim inf is 0 or lim sup is ~ a.e. So that a sequential probability ratio test terminates with probability one, except perhaps for one value of θ. There is no example known to showthat there may exist a θ₀for whichthe sequence density ratios has a positive lim inf and a finite lim sup.