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Martingale difference arrays and stochastic integrals
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  • Published: April 1986

Martingale difference arrays and stochastic integrals

  • Helmut Strasser1 

Probability Theory and Related Fields volume 72, pages 83–98 (1986)Cite this article

  • 212 Accesses

  • 16 Citations

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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. (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. (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. (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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Refrences

  1. Gaenssler, P., Haeusler, E.: On functional central limit theorems for martingales. Preprint, University of Munich (1984)

  2. Helland, I.S.: Central limit theorems for martingales with discrete continuous time. Scand. J. Statistics 9, 79–94 (1982)

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Authors and Affiliations

  1. Mathematical Institute, University of Bayreuth, PF 3008, D-8580, Bayreuth, Federal Republic of Germany

    Helmut Strasser

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  1. Helmut Strasser
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Cite this article

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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  • Received: 06 June 1984

  • Revised: 18 October 1985

  • Issue Date: April 1986

  • DOI: https://doi.org/10.1007/BF00343897

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Keywords

  • Likelihood Ratio
  • Limit Theorem
  • Taylor Expansion
  • Central Limit Theorem
  • Functional Limit
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