content_image/312619.jpg
ized random functions in G ∗ = D $^{∗}$, rather than S $^{∗}$. One can similarly define random generalized functions on other spaces of test functions, not necessar- ily infinitely differentiable, e.g. on D$_{k}$ of k times continuously differentiable functions with compact support, leading to space D ∗ k of random linear con- tinuous functionals on D$_{k}$. 

## 4.2 Stochastic convergence of random generalized func- tions 

A random sequence b$_{n}$ of elements of a space of generalized functions G ∗ con- verges to zero in probability: b$_{n}$ →$_{p}$ 0 in G $^{∗}$, or almost surely: b$_{n}$ →$_{a.s.}$ 0 in G ∗ if for any finite set ψ$_{1}$, ...ψ$_{v}$ ∈ G the random vectors (( b$_{n}$, ψ$_{1}$ ) , ... ( b$_{n}$, ψ$_{v}$ )) →$_{p}$ (0 , ..., 0) or correspondingly, Pr((( b$_{n}$, ψ$_{1}$ ) , ... ( b$_{n}$, ψ$_{v}$ )) → (0 , ..., 0)) = 1 . 

Similarly, convergence in distribution of generalized random processes b$_{n}$ ⇒$_{d}$ b is defined by the convergence of all multivariate distributions for ran- dom vectors (( b$_{n}$, ψ$_{1}$ ) , ... ( b$_{n}$, ψ$_{v}$ )) →$_{d}$ (( b, ψ$_{1}$ ) ... ( b, ψ$_{v}$ )) for any set ψ$_{1}$, ..., ψ$_{v}$ ∈ G. 

Remark 1. (a) If b$_{n}$ − b →$_{p}$ 0 in S ∗ then F t ( b$_{n}$ ) − F t ( b ) →$_{p}$ 0 in S ∗ and F t − $^{1}$( b$_{n}$ ) − F t − $^{1}$( b ) →$_{p}$ 0 in S $^{∗}$. Indeed, for any set ψ$_{1}$, ...ψ$_{v}$ ∈ S 

<img src='content_image/312619.jpg'>

and since the set F t ( ψ$_{1}$ ) , ..., F t ( ψ$_{v}$ ) ∈ S then (( b$_{n}$ − b, F t ( ψ$_{1}$ )) , ..., ( b$_{n}$ − 

