On $\mathbf{2\times2}$ determinants originating from survival probabilities in homogeneous discrete time risk model

DOI10.1007/S00025-022-01736-YzbMATH Open1507.60059arXiv2102.06987MaRDI QIDQ6360536

Publication date: 13 February 2021

Abstract: We analyze

2 i m e s 2

Hankel-like determinants

D_{n}

that arise in the initial values problem for the ultimate time survival probability

v a r p h i (u)

in a homogeneous discrete time risk model

W (n) = u + k a p p a n + s u m_{i = 1}^{n} Z_{i}

, where

Z_{i}

are positive integer valued i.i.d. random claims, the initial surplus

u i n m a t h b b N_{0}

and the income rate

k a p p a = 2

. We prove the asymptotic version of a recent conjecture on the non--vanishing and monotonicity of

D_{n}

and derive explicit formulas for the initial values

v a r p h i (0)

,

v a r p h i (1)

of a recurrence that yields survival probabilities. In cases when

Z_{i}

are Bernoulli or Geometrically distributed, the conjecture on

D_{n}

is shown to hold for all

n i n m a t h b b N_{0}

. Additionally, a generating function

X i (s)

for ultimate survival probabilities

v a r p h i (u)

is derived.

Mathematics Subject Classification ID

Sums of independent random variables; random walks (60G50) Risk models (general) (91B05) Actuarial mathematics (91G05)

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