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Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$ and let$C_{\infty}(\Phi')$ be the collection of all continuous mappings $x:[0,\infty)\rightarrow \Phi'$ equipped with the topology of local uniform convergence. Inthis paper we introduce sufficient conditions for tightness of probabilitymeasures on $C_{\infty}(\Phi')$ and for weak convergence on $C_{\infty}(\Phi')$for a sequence of $\Phi'$-valued processes.

Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space