From here according to the theorem about continuous correspondence between image and preimage of Laplace transformation (see [fel67]) we obtain that the distribution Fy(-| x) (dFy(u| x) = u fyyu}x du) converges weakly to the distribution Fo(-| x) (dFo(uI x) = uv(u,x) du). Lemma is proved.

Example 1

For inverse gamma process with parameters 7 and S we have v(t, x) = v(t, 0) = v(t):

x fM = SHW)e-' = = n^Tr, '-1'M"-' — ^ = vM (x — 0).

Example 2

For the process of records of standard Wiener process (see [har01]) we have:

Corollary 1. If conditions of lemma 1 are fulfilled, and the function fy (t| x) is differentiable with respect to t, then

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