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Math Help - uniformly integrable

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    uniformly integrable

    Hallo!
    Is the following sequence of integrable random variables (X_h)_{h \in [0,1]} also uniformly integrable?

    X_h=\prod\limits_{n=1}^{\frac{T}{h}}(1+\alpha_{(n-1)h}(\exp\{\mu_{(n-1)h}h+ \sigma (W_{nh}-W_{(n-1)h})\} -1))^\gamma
    whereas
    W is a standard Brownian motion,
    \sigma a constant  \in \matbb{R}_+,
    \alpha_t is a random variable with values in [0,1],
    \mu_t is a standard-normal distributed random variable and \mu_t is continous in t
    [0,T] is the time interval and it is required that N:=\frac{T}{h} \in \matbb{N} and
    \gamma is a constant \in (0,1)

    I also know that X_h converges in probability for h \to 0 to a integrable random variable X.
    I've no idea how to show it.
    Can anybody help me?

    I found out that \mathbb{E}[X_h]\leq \mathbb{E}[X]<\infty \; \forall h \in [0,1] \quad and  X_h converges in probability to this random variable  X.

    Can I now conclude that (X_h)_{h \in [0,1]} is uniformly integrable?
    Last edited by Juju; February 6th 2011 at 01:21 AM.
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