Abstract:
inthisnote, we obtain a chung's integral test for self-normalized sums of i.i.d. random variables. furthermore, we obtain a convergence rate of chung law of the iterated logarithm for self-normalized sums.

Abstract:
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(\lambda)=\exp{\lambda A_t-\lambda ^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq 0}A_t/{B_t(\log \log B_{t})^{1/2}} and variants thereof, when EY_t(\lambda )\leq 1 or when Y_t(\lambda) is a supermartingale, for all \lambda belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with A_t=M_t and B_t=\sqrt < M>_t, and sums of conditionally symmetric variables d_i with A_t=\sum_{i=1}^td_i and B_t=\sqrt\sum_{i=1}^td_i^2. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in R^m, m\ge 1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving \sum_{i=1}^td_i and \sum_{i=1}^td_i^2.

Abstract:
We prove a new Donsker's invariance principle for independent and identically distributed random variables under the sub-linear expectation. As applications, the small deviations and Chung's law of the iterated logarithm are obtained.

Abstract:
The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation.

Abstract:
We prove an Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. As many other game-theoretic proofs, our proof is self-contained and explicit.

Abstract:
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments. For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sub-linear expectation for random variables with only finite variances.

Abstract:
In this paper, motivated by the notion of independent identically distributed (IID) random variables under sub-linear expectations initiated by Peng, we investigate a law of the iterated logarithm for capacities. It turns out that our theorem is a natural extension of the Kolmogorov and the Hartman-Wintner laws of the iterated logarithm.

Abstract:
We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary part of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and technics from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.

Abstract:
In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norming in the spirit of the classical Law of Iterated Logarithm. Key words: Martingales, exponential estimations, moment, Banach spaces of random variables, tail of distribution, conditional expectation.