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STS518 B.H. Jasiulis G. et al.
introduced in [6]) with ( ) = λ є assuming that = m is finite

1
α
1
+

or the truncated – moment H(t) ∶= ∫ () is regularly varying.
[0,)
Theorem 2. Let { ∶ є } be a Kendall random walk with parameter > 0,

unit step distribution ( ) = є + and G(x) ∶= ∫ [0,∞) (1 − − ) +
1
()
(i) If [ ] = < ∞, then as → ∞,

1

where the cdf of random variable X is given by

(ii) Suppose that H є RVθ where 0 ≤ θ < α. Then there exists an increasing
function U(x) such that U(1/(1-G(x)) ⁓ x and

where has distribution, which is a convex linear combination of an
exponential and a gamma distribution

a
where (, ) denotes the measure with the density b /
Γ(a) −1 exp{−bx}1 [0,∞) (u).

Proof.

(i) Let F denotes the cdf of the unit step . First notice that H( 1/ /x) →
1
m and F( 1/ /x) → 1 , as → ∞. Since the Williamson transform for
α
−1/ is given by the following formula (see [1]):

α
n
then we obtain G( 1/ /x) → exp{−m x } as → ∞. To complete the
α
proof it suffices to check that the limiting measure has exactly the Williamson
transform exp{− x }.
α

(ii) The second part of theorem can be proved using limit theorem for
renewal process N(t) ∶= inf { n: +1 > t} constructed by the Kendall
convolution (see Theorem 6 in [10]). We use the result that

Since − > 0, 1/(1 − ()) is asymptotically equal to a strictly increasing
function V(x) є R − (see [4], Section 1.5.2, Theorem 1.5.4, p.23) and

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