ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Consider MDAs (X ni) and (Y ni), and stopping times τ n (t), 0≦t≦1. Denote $$S_n (t) = a_0 + \sum\limits_{i = 1}^{\tau _n (t)} {X_{ni,} {\text{ }}T_n (t) = b_0 + \sum\limits_{i = 1}^{\tau _n (t)} {Y_{ni,} } }$$ and let ϕ: ℝ→ℝ be a function. If the common distribution converges and if S t , T t denote the corresponding limiting processes then we give conditions such that the martingale transforms $$\sum\limits_{i = 1}^{\tau _n (t)} \varphi (S_{n,i - 1} )Y_{ni}$$ converge weakly to the stochastic integral $$\int\limits_0^t {\varphi (S)dT.}$$ This result has important consequences for functional central limit theorems: (1) If the MDAs are connected by a difference equation of the form $$X_{_{ni} } = \varphi \left( {S_{_{n,i - 1} } } \right)Y_{_{ni,} } $$ , then weak convergence of T n (t) implies that of S n (t), and the limit satisfies the stochastic differential equation $$dS = \varphi \left( {S_{} } \right)d T.$$ . This observation leads to functional limit theorems for diffusion approximations. E.g. we obtain easily a result of Lindvall, [4], on the diffusion approximation of branching processes. (2) If the MDA (X ni ) arises from a likelihood ratio martingale then the limit satisfies $$S_t = 1 + \int\limits_0^t {SdT,}$$ which leads to the representation of the limiting likelihood ratios as exponential martingale: $$S_t = \exp (T_t - \frac{1}{2}[T,T]_t ).$$ This approximation by an exponential martingale has been proved previously by Swensen, [9], using a Taylor expansion of the log-likelihood ratio. (3) As a consequence we obtain a general functional central limit theorem: If $$\left( {\sum\limits_{i = 1}^{\tau _n (t)} {X_{_{ni} }^2 } } \right)$$ converges weakly to ([S, S] t ), then $$\left( {\sum\limits_{i = 1}^{\tau _n (t)} {X_{_{ni} }^{} } } \right)$$ converges weakly to (S t ), provided that the distribution of (S t ) is uniquely determined by that of ([S, S] t ). This assertion embraces previous central limit theorems, dealing with cases where the increasing process ([S, S] t ) is deterministic.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00343897
Permalink