Rukhin et al. (2010) proposed the non-overlapping template matching test as one of methods for statistical testing of randomness in cryptographic applications. This test is the very interesting, but statistical properties of this test and any methods on setting the template have not been shown. Our new contribution in this paper is to propose a modified version of this test including the setting of the template and to show how this modified test works effectively by some simulation studies.

Embedding a discrete-time autoregressive moving average (DARMA) process in a continuous-time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous-time and related discrete-time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases. Copyright 2007 The Author Journal compilation 2007 Blackwell Publishing Ltd.

When we use the estimators, obtained by solving Yule-Walker equations, of the coefficients of an autoregressive process, we cannot discriminate X, and Yo where all the solutions of the associated polynomial equation of X, are less than 1 in the absolute value and, at least, one of the solutions of that of Y, is greater than 1 in the absolute value. To discriminate between X, and Y,, Rosenblatt proposed a method. We propose another method by using a higher order monient.

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We consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model. We evaluate the asymptotic value of the conditional quasi-likelihood function when the number of observations tends to infinity. We show what properties of the time series we can find by examining the behaviour of the conditional quasi-likelihood function, even when the time series does not necessarily satisfy the model and is not necessarily Gaussian.