Summary
Consider MDAs (X ni) and (Y ni), and stopping times τ n (t), 0≦t≦1. Denote
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
converge weakly to the stochastic integral
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.
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(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.
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(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.
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Strasser, H. Martingale difference arrays and stochastic integrals. Probab. Th. Rel. Fields 72, 83–98 (1986). https://doi.org/10.1007/BF00343897
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DOI: https://doi.org/10.1007/BF00343897