A.s. convergence rate for Mandelbrot’s cascade in a random environment

Abstract.

For Mandelbrot’s cascade (Y_n) in an independent and identically distributed (i.i.d.) random environment ξ, we are interested in the a.s. convergence rate of the Mandelbrot’s martingale (W_n) to its limit W, where $W_n=Y_n/E_{\xi }{Y}_n$ is the normalized partition function. We obtain sufficient conditions under which W−W_n has an exponential convergence rate: $W-W_n=o(e^{-na})$ a.s. for some a > 0 explicitly calculated; we also find conditions under which W−W_n has a polynomial convergence rate: $W-W_n=o(n^{-\alpha })$ a.s. for some α > 0. Similar conclusions hold for Mandelbrot’s cascade in a varying environment.

Keywords:

• A
• convergence rate
• Mandelbrot’s cascade
• random environment
• varying environment

Acknowledgments

The authors are grateful to anonymous referees and Professor Quansheng Liu for their very valuable comments and remarks, which significantly contributed to improving the quality of the article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (Grants no. 12271062), by Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No. 2018MMAEZD02).