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It is well-known that the Kolmogorov SLLN (non-i.i.d. case) fails for pair-wise independent random variables. However, as shown in the paper it can be saved even for orthogonal random variables if one allows permutations. We prove it in the setup of Banach space valued random variables.

Rademacher sums of exchangeable finite system of Banach space valued random variables are considered. The equivalence of the tails of the related distributions is established. The results seem to be new also for scalar random variables.

The main aim of this paper is to prove a bilateral inequality for the Rademacher functions. As a corollary we give a maximal inequality for exchangeable random variables that recently has been published in [Pruss, Proc. Amer. Math. Soc. 126: 1811–1819, 1998]

We find conditions on a sequence of random variables to satisfy the SLLN under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.

We use our maximum inequality for p-th order random variables (p>1) to prove a strong law of large numbers (SLLN) for sequences of p-th order random variables. It was known that the above condition, under the additional assumption, of monotonicity, implies SLLN (Erdos (1949), Gal and Koksma (1950), Gaposhkin (1977), Moricz (1977)). Getting rid of the monotonicity condition, the inequality enables to extend the general result to p-th order random variables, as well as to the case of Banach-space-valued random variables.

The goal of the paper is twofold: (1) to show that the exact value in the Meńshov-Rademacher inequality equals 4/3, and (2) to give a new proof of the Meńshov-Rademacher inequality by use of a recurrence relation

For any sequence of random variables, maximal inequalities from which we can derive conditions for the a.s. convergence to zero of the normalized differences is obtained. In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN; this condition is an improvement on the well-known Móricz conditions.

For any sequence of random variables, maximal inequalities from which we can derive conditions for the a.s. convergence to zero of the normalized differences is obtained. In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN; this condition is an improvement on the well-known Móricz conditions.

An analogue of the fact of convergence of the series for all signs for the convergenc for only almost all signs is found. An application of the result to the Nikishin problem is considered

We show that in any two-dimensional normed space there exists a collection of vectors such that the greedy algorithm fails to be optimal. 

A maximum inequality on rearrangement of summands of elements of a normed space is proved. The inequality is unimprovable and generalizes well-known results due to Garsia, Maurey and Pisier, Kashin and Saakyan, Chobanyan and Salehi, and Levental

Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin's well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.

A maximum inequality on rearrangement of the collection of elements of a normed space is stated. The inequality is unimprovable and generalizes well-known results due to Garsia, Maurey and Pisier, Kashin and Saakyan, Chobanyan and Salehi, and Levental

We find a sufficient condition for a.s. convergence to zero of summands given that a sum of two sequences of random variables a.s. converges to zero. The condition turns out to be weaker than that used in the monograph by Loeve and in a paper by Martikainen. Our result is also used in construction of a counter-example regarding the a.s. convergence of a rearranged series.

One variant of the transference lemma is proved. By its application we get a refined version of the Garsia inequality for orthogonal systems. Moreover, it is shown that the fulfillment of the “sigma-theta” condition for the Fourier series of a continuous periodic Banach space valued function implies the uniform convergence of a rearranged series to the function

We prove the Kahane contraction principle for the tail probabilities of linear combinations of a finite exchangeable system of random variables. Our note goes back to the maximum inequalities for permutations developed by Steinitz, Garsia, Nikishin, Maurey and Pisier, and Kashin having applications in analysis and function theory. The result seems to be new, even in the case of reals.

It is shown that every infinite-dimensional real Banach space  contains a sequence, were its subsequence satisfies some editional properties. This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

The interrelation between signs and permutations in maximum inequalities is studied in this paper. The relationship is based on a lemma that reduces a rearrangement problem to a problem of choosing signs. It helps simplify proofs and find new facts and general settings. Obtained inequality is unimprovable (the inverse inequality also holds with some other constant); it generalizes well-known results due to Garsia; Maurey, and Pisier; Kashin and Saakyan; Chobanyan and Salehi; and Levental. All the inequalities of this paper can be restated as maximum inequalities for exchangeable random variables.

We give a transference maximum inequality establishing interrelation between the sign-convergence and the rearrangement convergence of functional series. We use this inequality to get a generalization of the Maurey and Pisier sign-permutation relationship and the general Garsia and Nikishin type local inequalities, as well as the Nikishin and Garsia convergence theorems on the almost sure convergence of functional series. We also get some maximum inequalities that might be useful in the analysis of the Kolmogorov Rearrangement Conjecture on the systems of convergence of orthogonal series as well as Garsia’s local form of the conjecture

It is shown that the rearrangement theorem is not true for infinite-dimensional Banach spaces.