A Proof of the Burkholder Theorem For Martingale Transforms
T. Shintani · Proceedings of the American Mathematical Society · 1981
If $g$ is the transform of an ${L^1}$-bounded martingale $f$ under a predictable sequence $\upsilon$ satisfying ${\text {sup}_n}\left | {{\upsilon _n}} \right | < \infty$ almost everywhere, then a proof of the convergence of $g$ is given using an approximation of $f$ by a martingale of bounded variation.