# Quadratic Truncated Random Return in Distributional LQR: Positive Definiteness, Density, and Log-Concavity

arXiv: https://arxiv.org/abs/2510.23332 · HTML: https://mlsanity.com/paper/2510.23332/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 6 of 6)

- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Distributional Reinforcement Learning with Quantile Regression](https://mlsanity.com/paper/1710.10044/) (2017)
- [Risk-Constrained Reinforcement Learning with Percentile Risk Criteria](https://mlsanity.com/paper/1512.01629/) (2015)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Policy Evaluation in Distributional LQR (Extended Version)](https://mlsanity.com/paper/2401.10240/) (2023)

## Abstract (arXiv)

Distributional linear quadratic regulator (LQR) is a new framework that integrates the distributional reinforcement learning and classical LQR, which offers a new way to study the random return instead of the expected cost. Unlike iterative approximation using dynamic programming in the DRL, a closed-form expression for the random return can be exactly characterized in the distributional LQR, which is defined over infinitely many random variables. In recent work [1, 2], it has been shown that this random return can be well approximated by a finite number of random variables, which we call truncated random return. In this paper, we study the truncated random return in the distributional LQR. We show that the truncated random return can be naturally expressed in the quadratic form. We develop a sufficient condition for the positive definiteness of the block symmetric matrix in the quadratic form and provide the lower and upper bounds on the eigenvalues of this matrix. We further show that in this case, the truncated random return follows a positively weighted non-central chi-square distribution if the random disturbances admits Gaussian, and its cumulative distribution function is log-concave if the probability density function of the random disturbances is log-concave.