Hallo!

Is the following sequence of integrable random variables also uniformly integrable?

whereas

is a standard Brownian motion,

a constant

is a random variable with values in [0,1],

is a standard-normal distributed random variable and is continous in

[0,T] is the time interval and it is required that and

is a constant

I also know that converges in probability for to a integrable random variable .

I've no idea how to show it.

Can anybody help me?

I found out that and converges in probability to this random variable

Can I now conclude that is uniformly integrable?