We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant $K > 0$ such that if $\mathcal{M}$ is a semi-finite von Neumann algebra and $( \mathcal{M}_n )^{ \infty }_{n = 1}$ is an increasing filtration of von Neumann subalgebras of $\mathcal{M}$ then for any given martingale $x = ( x_n )^{\infty}_{n = 1}$ that is bounded in $L^2 ( \mathcal{M} ) \cap L^1 ( \mathcal{M} )$, adapted to $( \mathcal{M}_n )^{\infty}_{n = 1}$, there exist two martingale difference sequences, $a = ( a_n )_{n = 1}^\infty$ and $b = ( b_n )_{n = 1}^\infty$, with $dx_n = a_n + b_n$ for every $n \geq 1$,

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{2} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{2} \leq 2 \| x \|_2,

and

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{1, \infty} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{1, \infty} \leq K \| x \|_1.

As an application, we obtain the optimal orders of growth for the constants involved in the Pisier–Xu non-commutative analogue of the classical Burkholder–Gundy inequalities.

We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section 2 the equivalence between Shimogaki’s theorem and some martingale inequalities will be established, and in Section 3 the equivalence between Boyd’s theorem and martingale inequalities with change of probability measure will be established.

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.

Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.