# uniformly integrable

#### Juju

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

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

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