# Thread:

1. ## 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?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9160608649253845, "perplexity_flag": "head"}