content_image/304670.jpg content_image/304667.jpg content_image/304669.jpg content_image/304668.jpg content_image/304666.jpg content_image/304665.jpg
Proof: As mentioned in Lemma 1, the topology of weak convergence on P is metrizable with the L´evy-Prohorov metric [15] and defined as 

<img src='content_image/304670.jpg'>

for any µ, ν ∈ P , where A ( δ ) denotes the set of all x ∈$_{R}$ which lie a d -distance less than δ from A . 

Similarly as in the proof of Lemma 1, it suffices to show that Λ$_{0}$( γ, q$_{0}$ ) is both tight and closed in P . The tightness can be shown by the same arguments as in Lemma 1. In the following, we prove that Λ$_{0}$( γ, q$_{0}$ ) is closed in P . 

Let B$_{m}$ = $^{[}$− 1 $_{m}$, 1 m ] for m = 1 , 2 , . . . . Let { µ$_{n}$ } ∞ n =1 be a convergent sequence in Λ$_{0}$( γ, q$_{0}$ ) with limit µ$_{0}$ . For any m ∈$_{N}$ , there exists an n$_{m}$ such that L ( µ$_{n}$, µ$_{0}$ ) < 1 m for all n > n$_{m}$ . By the definition of L in (125), we have for any m ∈$_{N}$ and n > n$_{m}$ , 

and 

For any n ∈$_{N}$ $^{⋃}${ 0 } , we have 

<img src='content_image/304667.jpg'>

<img src='content_image/304669.jpg'>

<img src='content_image/304668.jpg'>

so for any m ∈$_{N}$ , there exists an n ′ m such that µ$_{n}$ ( B$_{m}$ ) ≤ µ$_{n}$ ( { 0 } ) + 1 $_{m}$. Therefore, according to (126) and (127), for all m ∈$_{N}$ and n > max { n$_{m}$, n $^{′}$$_{m}$} , 

Thus we have µ$_{0}$ ( { 0 } ) = q$_{0}$ by letting m → ∞ . 

<img src='content_image/304666.jpg'>

Moreover, let f ( x ) = x 2 which is continuous and bounded below. By weak convergence [15, Sec- tion 3 . 1 ], we have 

<img src='content_image/304665.jpg'>

