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Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$: $$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\phi_n(t)\xi_n(\omega),$$where $\mu_X(t)=\mathbb{E}[X(t)]$, $\{\xi_n\}_{n=1}^\infty$ are pairwise uncorrelated random variables with $\mathbb{E}[\xi_n]=0$ and $\mathbb{V}[\xi_n]=1$, and $\{(\nu_n,\phi_n)\}_{n=1}^\infty$ correspond to the non-zero eigenvalues and eigenfunctions of the operator $$ \mathcal{C}:L^2([0,T])\rightarrow L^2([0,T]),\;\;\mathcal{C} f(t)=\int_0^T \text{Cov}(X(t),X(s))\,f(s)\,ds. $$ Denote the partial sum by $$ X_N(t,\omega)=\mu_X(t)+\sum_{n=1}^N \sqrt{\nu_n}\phi_n(t)\xi_n(\omega).$$ The sequence $\{X_N\}_{N=1}^\infty$ converges in $L^2([0,T]\times\Omega)$. Moreover, if $(t,s)\mapsto \text{Cov}(X(t),X(s))$ is continuous, the sum converges in the sense of $\sup_{t\in [0,T]} \mathbb{E}[(X_N(t)-X(t))^2]\rightarrow0$.

My two questions are:

  1. Under which assumptions on $X$ do we have that the sequence $\{X_N\}_{N=1}^\infty$ is uniformly bounded on $L^\infty([0,T]\times\Omega)$? That is, there exists a $C>0$ such that $\|X_N\|_{L^\infty([0,T]\times\Omega)}\leq C$ for all $N$.

  2. Under which assumptions on $X$ do we have that the sequence $\{X_N\}_{N=1}^\infty$ converges on $L^\infty([0,T]\times\Omega)$ to $X$? That is, $\lim_N \|X_N-X\|_{L^\infty([0,T]\times\Omega)}=0$.

  3. Under which assumptions on $X$ do we have that the sequence $\{X_N\}_{N=1}^\infty$ converges on $L^p([0,T]\times\Omega)$ to $X$, $1\leq p<\infty$? That is, $\lim_N \|X_N-X\|_{L^p([0,T]\times\Omega)}=0$.

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