Page 171 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya
Theorem 2 Let the condition

(3)

be valid. If λ ↓ 0, then the following asymptotic equalities take place:

where , ∈ , are some positive constants.

The next theorem, see (Yarovaya, 2017c), is valid under the condition of
infinite variance of underlying random walk jumps.
Theorem 3 Let the condition

(4)

be valid. If λ ↓ 0, then there are the following asymptotic equalities:

where , is some positive constant for each dimension d of the lattice ℤ .
Let βi > 0 for = 1, 2, . . . , and the operator ℋ has a finite
1 , 2 …,
number of positive eigenvalues. We denote the largest of them by λ0, and the
corresponding normalized vector by . Then for all ∈ and t → ∞, see
(Khristolyubov and Yarovaya, 2019), the limit statements hold:

where

For = 0 we have

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