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In this paper, we define and study a first order moving average model with Laplace marginal distributions. Extensions to higher orders are discussed. A first order moving average process with mixed Laplace distributions as marginal is developed and studied. The model helps us to simulate mixed Laplace distribution with negative weights. We also introduce a first order moving average process as the mixture of asymmetric Laplace marginals.

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This paper we introduced a new distribution namely the multivariate Esscher transformed Laplace distribution. Various properties of the distribution are studied and the applications are discussed. Further we develop an autoregressive process with multivariate ETL marginal and study its properties. A Levy process based on this multivariate infinitely divisible distribution is known as Laplace motion, and its marginal distributions are multivariate generalized Esscher transformed Laplace distribution.

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