Start your brilliant career with a degree from Australia's #1 ranked university.


Find contact details for any general enquiries.


Find contact details for any general enquiries.

Study with us

Our graduates gain the knowledge and skills to lead organisations, develop public policy, create new companies and undertake research.

Our research

Our research is focused on issues that are highly significant for organisations, the Australian economy, and society at large.

Student resources

Whether you're a new or continuing student, you can find everything you need here about managing your program and the opportunities available to you.


Our alumni may be found in the world’s leading companies, policy agencies and universities.

Contact us

Find contact details for any general enquiries.

Distributional Representations and Dominance of a Levy Process over its Maximal Jump Processes

Distributional Representations and Dominance of a Levy Process over its Maximal Jump Processes

Boris Buchmann, Yuguang Fan and Ross Maller

Distributional identities for a L evy process X, its quadratic variation process V, and its maximal jump processes sup 0<s \le t DX_s, are derived, and used to make (as t ->0) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of X.

Apart from providing insight into the connections between X, V , and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application we study a self-normalized versions of X_t, that is, X_t after division by sup0<s \le t DX_s, or by sup0<s \le t |DX_s|. Thus we obtain necessary and sufficient conditions for X_t/ sup0<s\le t DX_s and X_t/sup 0<s\le t |DX_s| to converge in probability to 1, or infinity, as t ->0, so that X is either comparable to, or dominates, its largest jump.

The former situation tends to occur when the singularity at 0 of the L evy measure of X is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of X at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous large time (as t to infinity) versions of the results can also be obtained.

Partnership opportunities available

Our College is actively engaged in partnering with industry for the co-creation of value in areas congruent with our research agenda

Find out more >>

Other research you might be interested in


Updated:   8 March 2019 / Responsible Officer:  CBE Communications and Outreach / Page Contact:  College Web Team