There are so many possible answers and the subject is so large!

When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

• Consistency under Sampling of Exponential Random Graph Models

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.

Edit 1: If the question is about which parametric families are closed under taking mixtures, then the following references are relevant

• Misiewicz et. al. (2005). Classes of measures closed under mixing and convolution. Weak stability
• Teicher (1960). On the Mixture of Distributions

There are so many possible answers and the subject is so large!

When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

• Consistency under Sampling of Exponential Random Graph Models

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.

Edit 1: If the question is about which parametric families are closed under taking mixtures, then the following references are relevant

• Misiewicz et. al. (2005). Classes of measures closed under mixing and convolution. Weak stability
• Teicher (1960). On the Mixture of Distributions

There are so many possible answers and the subject is so large!

When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

• Consistency under Sampling of Exponential Random Graph Models

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.

There are so many possible answers and the subject is so large!

When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

• Consistency under Sampling of Exponential Random Graph Models

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.

Edit 1: If the question is about which parametric families are closed under taking mixtures, then the following references are relevant

• Misiewicz et. al. (2005). Classes of measures closed under mixing and convolution. Weak stability
• Teicher (1960). On the Mixture of Distributions