# Stopping Times and Tightness. II

@article{Aldous1978StoppingTA, title={Stopping Times and Tightness. II}, author={David J. Aldous}, journal={Annals of Probability}, year={1978}, volume={17}, pages={586-595} }

To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of finite-dimensional distributions. Thus in many settings, weak convergence to a continuous limit process can be deduced almost immediately from convergence of finite-dimensional distributions. These results may be technically useful in simplifying proofs of weak convergence, particularly in infinite…

## 363 Citations

### Equivalent conditions for the tightness of a sequence of continuous Hilbert valued martingales

- MathematicsNagoya Mathematical Journal
- 1987

In D. Aldous gave a sufficient condition for the tightness of a sequence (Xn)n≥0 of right continuous (with left limits) processes taking their values in a separable complete metric space S. As…

### Central Limit Theorems for Martingales with Discrete or Continuous Time

- Mathematics
- 1982

This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of…

### Weak convergence of sequences of semimartingales with applications to multitype branching processes

- MathematicsAdvances in Applied Probability
- 1986

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of…

### Convergence of values in optimal stopping and convergence of optimal stopping times

- Mathematics
- 2005

Under the hypothesis of convergence in probability of a sequence of cadlag processes $(X^n)$ to a cadlag process $X$, we are interested in the convergence of corresponding values in optimal stopping…

### Remarks on the functional central limit theorem for martingales

- Mathematics
- 1979

SummaryIt is shown that filling a gap in the proof of Lemma 3 in Rootzén (1977) and using an appropriate truncation procedure one obtains general necessary conditions for the functional CLT for…

### Functional limit theorems for Volterra processes and applications to homogenization

- MathematicsNonlinearity
- 2022

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process (yt)t⩾0 in the rough path topology. As an application, we establish weak convergence as ɛ → 0 of…

### A Criterion of Convergence of Generalized Processes and an Application to a Supercritical Branching Particle System

- MathematicsCanadian Journal of Mathematics
- 1991

The problem of convergence in distribution of a large class of generalized semimartingales to a continuous process is considerably simplified by a recent theorem of Aldous [1], in conjunction with a…

### Weak convergence of jump processes

- Mathematics
- 1992

This paper gives a necessary and sufficient condition for the weak convergence X n ⇒ X of general jump processes defined on R+, for Skorokhod topology, in terms of their predictable characteristics v…

### Central limit theorems for local martingales

- Mathematics
- 1980

This paper is involved with the following problem. Given a sequence of local martingales, say (Mn), under which conditions on the quadratic variations ([M,]), can we state the convergence in…

### ON WEAK CONVERGENCE OF SEMIMARTINGALES TO STOCHASTICALLY CONTINUOUS PROCESSES WITH INDEPENDENT AND CONDITIONALLY INDEPENDENT INCREMENTS

- Mathematics
- 1983

The authors study weak convergence of a sequence of semimartingales to an arbitrary stochastically continuous process independent or conditionally independent increments. The "semimartingale scheme"…

## References

SHOWING 1-8 OF 8 REFERENCES

### A limit theorem for measurable random processes and its applications

- Mathematics
- 1976

Let the measurable random processes (t), ..., (t),... and ((t) be defined on [0, 1]. There exists C such that for all n and t we have EI4,(t)IP 1. The following assertion is valid: if for any finite…

### Convergence of Probability Measures

- MathematicsThe Mathematical Gazette
- 1970

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

### Stopping times and tightness

- Ann . Probab .
- 1978

### Two topics in probability theory

- 1977