Difference between GMM and HMM

From what I understand:

• GMM is a probabilistic model which can model N sub population normally distributed. Each component in GMM is a Gaussian distribution.
• HMM is a statistical Markov model with hidden states. When the data is continuous, each hidden state is modeled as Gaussian distribution.

If these two informations are correct, what is the difference between GMM and HMM ?

Also, in time series case (continuous data), each state is one and only one Gaussian distribution ? there is no emission probability matrix ?

Thanks for your help !!! :)

Those two pieces of information are NOT correct. First, you have to understand the 'process' versus random variable (RV). HMM, even its name is 'model' actually it is a (stochastic) process that RV(s) change over time index t. GMM is usually used for the emission of the process (HMM). Comparing HMM to GMM is not apple to apple, one HMM is a stochastic process even its name is a model. and GMM is related to the distribution of random variables. HMM need time index (t or n) while GMM doesn't.

Q: Also, in time series case (continuous data), each state is one and only one Gaussian distribution? there is no emission probability matrix? A: This question is not understandable