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arXiv · 2305.06766

On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space

Partiswari Maharana Sabita Sahoo
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Abstract

In the present paper, the random series $\sum\limits_{m=0}^\infty c_m C_m(\varpi)q_m^{(ζ,η)}(u)$ in orthogonal Jacobi polynomials $q_m^{(ζ,η)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobi coefficients of a function in the weighted space $\mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}),\mathrm{p}>1.$ The random variables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients of symmetric stable process $Y_{ζ,η}(v,\varpi)$ of index $χ\in [1,2]$ for $ζ,η\geq 0,$ which are not independent. We prove that, under certain conditions on $\mathrm{p},ζ$ and $η,$ the random Fourier--Jacobi series converges in probability to the stochastic integral \begin{equation*} \int_{-1}^1 \mathfrak{g}(u,v)dY_{ζ,η}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g} \in \mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}).$

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Partiswari Maharana Sabita Sahoo. 2024-08-13. On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space. https://arxiv.org/abs/2305.06766

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