Numerical Analysis of Generalized Statistical Manifolds Associated with Stochastic Processes

1.A generalized statistical manifold is constructed for （time-） discrete stochastic processes characterized by n parameters 6=（θ_1···θ_n）。根据几何信息，在离散时间N时引入了n参数家族S_N ={p（X， θ， N） θ∈Ω}的概率密度函数p（X，θ， N）的随机变量X（N）。As the discrete time goesby， a time series of statistical manifolds··， S_1， S_2，···，S_N···is generated， andgathering them leads to a stratified statistical manifold， which is the direct product space S×Z of a statistical manifold S and a set Z of integers。关于一个广义的统计矩阵，一个扩展的连接连接的两层是在牛顿-Cartan理论的牛顿重力理论和一个元素粒子物理学中的希格斯场被格式化为扩展的沟场。然后，扩展的曲线张力器是由扩展的连接和α-连接构造的。将这个新的几何学方法应用于随机行走模型，我们发现它的通用统计矩阵具有双重结构.研究了利曼标量曲线R的物理作用。Janyszek和Mrugala （JM），解释了tfa对系统的可持续性的衡量标准. 3.研究物理学中的曲线作用，我们也构建了热同步系统或阻塞过程的新模型。有三种宝石被发现了，但这些模型的几何特征仍然像未来一样有效。

1.A generalized statistical manifold is constructed for (time-) discrete stochastic processes characterized by n parameters 6=(θ_1・・・θ_n). According to information geometry, a metric tensor and an α-connection are introduced on an n-parameter family S_N={p(X, θ, N) θ∈Ω} of probability density functions p(X, θ, N) for a random variable X(N) at a discrete time N. As the discrete time goesby, a time series of statistical manifolds・・・, S_1, S_2, ・・・, S_N・・・ is generated, andgathering them leads to a stratified statistical manifold, which is the direct product space S×Z of a statistical manifold S and a set Z of integers. On the generalized statistical manifold an extended connection connecting two layers is introduced after the Newton-Cartan theory of Newton gravity and a theory in elementary particle physics in which Higgs fields are formulated as extended gauge fields. And then extended curvature tensors are constructed from the extended connection and the α-connection. Applying this new geometrical method to a random walk model, we find that its generalized statistical manifold has a duality structure.2.The physical role of the Riemann scalar curvature R of thermodynamic equilibrium systems is studied. Janyszek and Mrugala (JM) interpreted the tfas a measure of the instability of the systems.3.To examine wider roles of curvatures in physics, we also construct new models of thermodynamic systems or stochastic processes. Three germs are obtained, but geometrical characteristics in these models are left as future works.