# Exponential Strong Converse for Successive Refinement with Causal Decoder Side Information

## TL;DR

This paper establishes an exponential strong converse theorem for the $k$-user successive refinement problem with causal decoder side information, showing that outside the rate-distortion region, the error probability approaches one exponentially fast.

## Contribution

It extends the strong converse theorem to the $k$-user case with causal side information, generalizing previous two-user results and unifying special cases.

## Key findings

- Exponential decay of success probability outside the rate-distortion region.
- Extension of the strong converse to the $k$-user successive refinement problem.
- Corollary for the case of $k=1$, recovering the El Gamal and Weissman result.

## Abstract

We consider the $k$-user successive refinement problem with causal decoder side information and derive an exponential strong converse theorem. The rate-distortion region for the problem can be derived as a straightforward extension of the two-user case by Maor and Merhav (2008). We show that for any rate-distortion tuple outside the rate-distortion region of the $k$-user successive refinement problem with causal decoder side information, the joint excess-distortion probability approaches one exponentially fast. Our proof follows by judiciously adapting the recently proposed strong converse technique by Oohama using the information spectrum method, the variational form of the rate-distortion region and H\"older's inequality. The lossy source coding problem with causal decoder side information considered by El Gamal and Weissman is a special case ($k=1$) of the current problem. Therefore, the exponential strong converse theorem for the El Gamal and Weissman problem follows as a corollary of our result.