Lacunary Series, Nonlinear Functionals and Banach Space Structure

Abstract: In a previous paper \cite{BT} we studied the asymptotic behavior of $| \sum_{k=1}^N a_k X_{n_k}|<em>p$ for lacunary sequences $(X</em>{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem in the case $1\le p&lt;2$. In the present paper we extend this result for nonlinear functionals $f_k (a_1 X_{n_1}, \ldots, a_k X_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pełczynski type theorems in Orlicz spaces $L_ψ$.

• The paper establishes new criteria for subsequence equivalence in Lp and Orlicz spaces, extending the classical Kadec–Pełczyński theorem to nonlinear functionals.
• It introduces a uniform subsequence principle using an equicontinuity condition that bridges classical limit theorems with exchangeable sequence analyses.
• Results provide necessary and sufficient conditions for ℓ2 equivalence in lacunary sequences, offering refined insights into the geometric structure of Banach spaces.

Lacunary Series, Nonlinear Functionals, and Banach Space Structure

Overview

This paper undertakes a substantial extension of the classical theory regarding lacunary series, random variables, and the structure of Banach spaces. Specifically, it generalizes key asymptotic results for lacunary sequences of random variables in $L_p$ spaces to a broad class of nonlinear functionals and Orlicz spaces. The work incorporates stochastic limit theorems, Banach space geometry, and the interplay between weak convergence, limit random measures, and nonlinear functionals. Central outcomes include uniform versions of the subsequence principle in Aldous's sense and necessary and sufficient conditions for Kadec–Pełczyński-type decompositions in Orlicz spaces.

Main Results and Theoretical Contributions

Extension of Kadec–Pełczyński Alternatives

The classical Kadec–Pełczyński theorem asserts that, for $p>2$, every normalized weakly null sequence in $L_p$ spaces admits a subsequence equivalent to either the basis of $\ell_2$ or $\ell_p$. For $1\leq p<2$, this dichotomy becomes more delicate and was only fully resolved in prior work for certain cases. The present paper establishes new criteria for the asymptotic equivalence of subsequences in $L_p$ and, crucially, Orlicz spaces—thereby expanding the scope to a significantly wider class of Banach function spaces.

The authors extend known necessary and sufficient conditions for the first alternative in the Kadec–Pełczyński theorem ($1\leq p<2$). For a determining sequence $(X_n)$ of random variables in $L_p$ with $p>2$0 and $p>2$1 uniformly integrable, a subsequence $p>2$2 is equivalent to the basis of $p>2$3 if and only if the associated limit random measure $p>2$4 satisfies

$p>2$5

This characterizes the subspace geometry of $p>2$6 in terms of the random measures arising from subsequential distributions and their $p>2$7-th moments.

Uniform Subsequence Principle for Nonlinear Functionals

A core innovation is the extension of asymptotic equivalence from linear functionals (norms of partial sums) to general nonlinear, symmetric functionals $p>2$8. Under a technical, explicit equicontinuity condition (Eq. (a2) in the text), the authors prove a uniform, subsequential limit theorem: for any $p>2$9, there exists a subsequence such that for all $L_p$0 and weights $L_p$1,

$L_p$2

where the limit value $L_p$3 is defined by replacing the sequence with its limit exchangeable counterpart.

This uniformity is crucial in Banach space theory: it ensures that the “renormalized” subsequence mimics the extremal behavior of exchangeable (conditionally i.i.d.) sequences, thereby transferring classical limit theorems from the i.i.d.\ to dependent lacunary settings in both linear and nonlinear regimes.

Kadec–Pełczyński-Type Theorems in Orlicz Spaces

The paper advances the theory further by moving from $L_p$4 to Orlicz spaces $L_p$5, employing Young (Orlicz) functions to address general rearrangement-invariant norms. The result reads:

Given a normalized, weakly null, uniformly integrable determining sequence $L_p$6 in $L_p$7 where $L_p$8 grows between linear and $L_p$9-power, there exists a subsequence equivalent to $\ell_2$0 if and only if

$\ell_2$1

This bridges Banach lattice theory and probability, showing that the underlying structure for weakly null subspaces in these settings continues to parallel, but critically extend, the $\ell_2$2 case.

Technical Approach

The proofs integrate classical and modern probabilistic methods. The authors build upon Aldous's subsequence principle and employ the machinery of limit random measures, exchangeable sequences, and conditional independence (via de Finetti-type representations). The uniformity of convergence in nonlinear functionals is established through a nuanced application of Ranga Rao's lemma on uniform integrability and weak convergence in metric measure spaces.

Equicontinuity of the sequence of functionals (as captured in condition (a2)) is fundamental: it allows the induction argument to build subsequences respecting global uniform bounds over all arguments $\ell_2$3. For Orlicz spaces, detailed convexity and growth control for the Orlicz function yield sharp norm comparison estimates.

Numerical and Contradictory Findings

• The uniform subsequential principle established for nonlinear functionals is unconditionally sharp under the technical hypotheses and recovers all classical results for sums of independent variables as special cases.
• In Orlicz spaces with $\ell_2$4 for $\ell_2$5, the necessary and sufficient conditions for equivalence to $\ell_2$6 mirror those in $\ell_2$7, but the moment criterion involves the corresponding Orlicz power ($\ell_2$8 as the integrability space for $\ell_2$9 with respect to the limit measure).
• It is stressed that the results do not extend trivially to $\ell_p$0 or Orlicz functions with superquadratic growth ($\ell_p$1), reflecting the deep distinction in the interplay between type and cotype in Banach space geometry.

Implications and Future Directions

The paper's contributions clarify the geometric landscape of $\ell_p$2 and Orlicz spaces via probabilistic structures, providing tools that can be adapted to:

• The fine classification of subspaces and basic sequences in rearrangement-invariant Banach spaces.
• The analysis of nonlinear statistics of random series, with applications to empirical processes, random matrix theory, and convex geometry.
• Ongoing inquiries into the structure of unconditional bases and their analogs for nonlinear or polynomial functionals.

Future research could explore further generalizations—addressing the impact of different types of growth in the Orlicz function, or relaxing the uniform integrability assumptions, as well as applications to function spaces beyond the rearrangement-invariant setting. Another promising direction is the study of vector-valued lacunary series and operator-valued Orlicz spaces.

Conclusion

The paper establishes robust subsequential principles for both linear and nonlinear functionals of lacunary sequences in $\ell_p$3 and Orlicz spaces, synthesizing probabilistic limit theory and Banach space geometry. It places precise necessary and sufficient conditions for $\ell_p$4-type behavior in a broad range of function spaces and opens new avenues for the investigation of Banach subspace structure and random series beyond classical $\ell_p$5 settings.