Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components $D$ grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small-$D$ regime, the characteristic function undergoes a sharp bond-dimension collapse for $D \gtrsim 300$ components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of $N = 2^{30}$ frequency modes on standard hardware, far beyond the $N = 2^{24}$ ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.

翻译：独立随机变量加权和的特征函数在量子化张量列车（QTT，亦称矩阵乘积态MPS）表示中展现出低秩结构，使其完全非高斯概率分布得以实现指数级压缩。在变量独立性假设下，全局特征函数可分解为局部项的乘积。其QTT低秩结构源于连续模型的本征谱光滑性，或离散模型中随分量个数$D$增长而呈现的谱能量集中现象。我们以伯努利和对数正态随机变量的加权和为例进行验证：在前者中，尽管存在对抗性的小$D$不可压缩区域，当$D \gtrsim 300$个分量时特征函数会发生尖锐的键维度坍塌，从而实现多对数复杂度的时间与内存缩放；在后者中，该方法可在标准硬件上达到$N = 2^{30}$个频率模态的高分辨率离散化，远超稠密实现固有的$N = 2^{24}$上限。这些压缩表示可高效计算风险价值（VaR）和预期短缺（ES），支撑量化金融及更广泛领域中的实际应用。