Weak convergence and tightness of probability measures in an abstract Skorohod space

Becem Saidani; Raluca M. Balan

arxiv: 1907.10522 · v1 · pith:EM42CXZWnew · submitted 2019-07-24 · 🧮 math.PR

Weak convergence and tightness of probability measures in an abstract Skorohod space

Raluca M. Balan , Becem Saidani This is my paper

Pith reviewed 2026-05-24 16:45 UTC · model grok-4.3

classification 🧮 math.PR

keywords tightnessweak convergenceSkorohod spacecadlag functionsLevy motionprobability measures on function spacesD([0,1];D)

0 comments

The pith

Tightness criteria for probability measures on D([0,1];D) are obtained by characterizing its relatively compact subsets under a Whitt metric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the space of right-continuous functions with left limits from [0,1] into the Skorohod space D, equipped with the Skorohod-type distance from Whitt. It follows the Billingsley approach to characterize relatively compact subsets of this space and thereby supplies explicit tightness criteria for measures on it. One such criterion has already been applied to establish the existence of a D-valued alpha-stable Levy motion. Additional results give conditions for weak convergence of random elements in the space and for the existence of processes with paths in the space from their finite-dimensional distributions.

Core claim

The space D([0,1];D) is equipped with the Skorohod-type distance introduced in Whitt (1980). Relatively compact subsets are characterized by modulus-of-continuity conditions that extend the classical Billingsley criteria; the resulting tightness criteria for probability measures on the space are the central results, together with a criterion for weak convergence and a criterion for existence of a process with sample paths in D([0,1];D) based on finite-dimensional distributions.

What carries the argument

The space D([0,1];D) of J1-càdlàg functions from [0,1] to D, equipped with the Whitt Skorohod-type metric; the modulus conditions that characterize its relatively compact subsets.

If this is right

The tightness criterion already used to prove existence of a D-valued alpha-stable Levy motion can be reused for other D-valued processes.
Weak convergence of random elements in D([0,1];D) holds whenever the finite-dimensional distributions converge and the tightness criterion is satisfied.
Existence of a process with paths in D([0,1];D) follows from convergence of its finite-dimensional distributions together with the tightness modulus conditions.
The same compactness characterization yields criteria for relative compactness of sets of paths in the space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same modulus conditions could be checked directly on the finite-dimensional distributions of a candidate process to obtain existence without separate tightness arguments.
The construction supplies a template for iterating the Skorohod construction one more level, producing spaces of paths with values in D-valued cadlag processes.
The criteria remain available for any target space that itself admits a Skorohod metric and a Billingsley-style compactness theorem.

Load-bearing premise

The space is metrized by the Whitt Skorohod-type distance and its elements are required to be right-continuous with left limits in the J1 topology on D, so that the classical Billingsley compactness arguments apply directly.

What would settle it

A concrete sequence of probability measures on D([0,1];D) that meets all the stated modulus conditions yet fails to be tight, or a tight sequence that violates the modulus conditions.

read the original abstract

In this article, we introduce the space $D([0,1];D)$ of functions defined on $[0,1]$ with values in the Skorohod space $D$, which are right-continuous and have left limits with respect to the $J_1$ topology. This space is equipped with the Skorohod-type distance introduced in Whitt (1980). Following the classical approach of Billingsley (1968, 1999), we give several criteria for tightness of probability measures on this space, by characterizing the relatively compact subsets of this space. In particular, one of these criteria has been used in the recent article Balan and Saidani (2018) for proving the existence of a $D$-valued $\alpha$-stable L\'evy motion. Finally, we give a criterion for weak convergence of random elements in $D([0,1];D)$, and a criterion for the existence of a process with sample paths in $D([0,1];D)$ based on its finite-dimensional distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper sets up tightness criteria for measures on D([0,1];D) with Whitt's metric by adapting Billingsley's classical conditions; one criterion already saw use in their 2018 Levy motion paper.

read the letter

The main takeaway is that this paper defines D([0,1];D) as the space of cadlag functions from [0,1] into D (with the inner J1 topology) and equips it with Whitt's Skorohod-type distance. It then gives several criteria for tightness of probability measures by characterizing relatively compact subsets, plus a weak convergence criterion and one based on finite-dimensional distributions. One of the tightness criteria was already applied in the authors' 2018 paper to construct a D-valued alpha-stable Levy motion, which supplies an external check that the tools are workable in practice. The approach follows Billingsley (1968, 1999) directly, adapting the usual marginals-plus-modulus conditions to the nested setting. The stress-test note found no inconsistencies with completeness or measurability, so the extension appears routine but correctly executed. What the paper does well is make the criteria explicit and ready for applications in stochastic processes that take values in function spaces. The citation pattern is clean, building on Billingsley and Whitt without circular support. A minor soft spot is that the core extension is a straightforward product-space adaptation rather than a deep conceptual shift, so the novelty sits mainly in the specific space and the ready-to-use statements. This is for probabilists who already work with Skorohod topologies and need to prove tightness or weak convergence for processes with paths in D([0,1];D). A reader facing similar nested-space problems would find the explicit criteria useful. It deserves a serious referee because the claims rest on established methods, the external application adds credibility, and the technical details are worth checking in review. Recommendation: send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the space D([0,1];D) of functions from [0,1] to the Skorohod space D that are right-continuous with left limits under the J1 topology. The space is equipped with the Skorohod-type metric of Whitt (1980). Following Billingsley's classical approach, the paper characterizes relatively compact subsets to obtain several tightness criteria for probability measures on this space. It also supplies a criterion for weak convergence of random elements in D([0,1];D) and a criterion for existence of a process with paths in D([0,1];D) based on its finite-dimensional distributions. One of the tightness criteria is noted to have been applied in Balan and Saidani (2018) to construct a D-valued alpha-stable Levy motion.

Significance. If the stated characterizations hold, the work supplies a direct, routine extension of Billingsley-style tightness and weak-convergence results to an iterated Skorohod space. This is useful for constructing and studying processes taking values in D, such as Levy motions with jumps. The independent external application in Balan and Saidani (2018) provides concrete corroboration of utility. The approach builds parameter-free on established Polish-space theory without introducing new ad-hoc assumptions.

minor comments (3)

[Abstract] The abstract states that the criteria 'characterize the relatively compact subsets' but does not indicate the precise form of the modulus-of-continuity condition adapted to the inner J1 metric; a one-sentence summary of the key modulus in the introduction would improve readability.
The manuscript should explicitly confirm that (D([0,1];D), d) is a Polish space under Whitt's metric (or cite the relevant completeness and separability arguments from Whitt (1980) or Billingsley), as this is the load-bearing prerequisite for applying the classical tightness theorems.
Notation for the outer and inner Skorohod metrics is not always distinguished in the text; a short notational convention paragraph would prevent reader confusion when both appear in the same statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, for recognizing its significance as a direct extension of Billingsley's classical results, and for recommending minor revision. The independent application cited in Balan and Saidani (2018) is indeed a useful corroboration of the criteria's utility.

Circularity Check

0 steps flagged

Minor self-citation for application only; derivation self-contained on classical results

full rationale

The paper extends Billingsley's tightness criteria and Whitt's metric to the space D([0,1];D) via standard arguments on Polish spaces and modulus-of-continuity conditions. No equations reduce by construction to fitted inputs or self-definitions. The sole self-citation (to Balan-Saidani 2018) appears only as an external application of one derived criterion for Levy motion existence, supplying independent corroboration rather than load-bearing justification for the present claims. All load-bearing steps cite external sources (Billingsley 1968/1999, Whitt 1980) whose results are independent of this work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard properties of the Skorohod space D with J1 topology and the applicability of Billingsley's tightness method to the newly defined space; no free parameters or invented entities beyond the space definition itself.

axioms (2)

standard math The Skorohod space D is equipped with the J1 topology and the functions are right-continuous with left limits.
Invoked in the definition of D([0,1];D) in the abstract.
domain assumption The Skorohod-type distance from Whitt (1980) makes the space suitable for applying Billingsley's criteria.
Stated as the equipping metric for the new space.

invented entities (1)

D([0,1];D) no independent evidence
purpose: Space of functions from [0,1] to D that are right-continuous with left limits w.r.t. J1 topology.
Newly introduced space in the abstract.

pith-pipeline@v0.9.0 · 5715 in / 1515 out tokens · 27661 ms · 2026-05-24T16:45:56.743989+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Balan, R. M. and Saidani, B. (2019). Stable L´ evy motion with values in the Skorokhod space: construction and approximation. To appear in J. Theor. Probab. Preprint available on arXiv 1809.02103

work page internal anchor Pith review Pith/arXiv arXiv 2019
[2]

and Rosinski, J

Basse-O’Connor, A. and Rosinski, J. (2013). On the uniform con vergence of random series in Skorohod space and representation of c` adl` ag inﬁnitely d ivisible processes. Ann. Probab. 41, 4317-4341

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Billingsley, P. (1995). Probability and Measure. Third Edition. Wiley, New York

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Hult, H. and Lindskog, F. (2005). Extremal behaviour of regula rly varying stochastic processes. Stoch. Proc. Appl. 115, 249-274

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Skorokhod, A. V. (1956). Limit theorems for stochastic proc esses. Th. Probab. Appl. 1, 261-290

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Skorokhod, A. V. (1957). Limit theorems for stochastic proc esses with independent increments. Th. Probab. Appl. 2, 138-171

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Whitt, W. (1980). Some useful functions for functional limit th eorems. Math. Oper. Res. 5, 67-85. 21

work page 1980

[1] [1]

Balan, R. M. and Saidani, B. (2019). Stable L´ evy motion with values in the Skorokhod space: construction and approximation. To appear in J. Theor. Probab. Preprint available on arXiv 1809.02103

work page internal anchor Pith review Pith/arXiv arXiv 2019

[2] [2]

and Rosinski, J

Basse-O’Connor, A. and Rosinski, J. (2013). On the uniform con vergence of random series in Skorohod space and representation of c` adl` ag inﬁnitely d ivisible processes. Ann. Probab. 41, 4317-4341

work page 2013

[3] [3]

Billingsley, P. (1995). Probability and Measure. Third Edition. Wiley, New York

work page 1995

[4] [4]

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York

work page 1968

[5] [5]

Billingsley, P. (1999). Convergence of Probability Measures. Second Edition. Wiley, New York

work page 1999

[6] [6]

de Haan, L and Lin, T. (2001). On convergence toward an extre me value distribution in C[0; 1]. Ann. Probab. 29, 467-483

work page 2001

[7] [7]

Donsker, M. (1951). An invariance principle for certain probabilit y limit theorems. Mem. Amer. Math. Soc. 6

work page 1951

[8] [8]

Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Con- vergence. John Wiley, New York

work page 1986

[9] [9]

and Lindskog, F

Hult, H. and Lindskog, F. (2005). Extremal behaviour of regula rly varying stochastic processes. Stoch. Proc. Appl. 115, 249-274

work page 2005

[10] [10]

Kallenberg, O. (1983). Random measures. Third edition. Academic Press, London

work page 1983

[11] [11]

Skorokhod, A. V. (1956). Limit theorems for stochastic proc esses. Th. Probab. Appl. 1, 261-290

work page 1956

[12] [12]

Skorokhod, A. V. (1957). Limit theorems for stochastic proc esses with independent increments. Th. Probab. Appl. 2, 138-171

work page 1957

[13] [13]

Whitt, W. (1980). Some useful functions for functional limit th eorems. Math. Oper. Res. 5, 67-85. 21

work page 1980