Leaven
发表于 2025-3-23 09:57:13
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Eeg332
发表于 2025-3-23 15:32:41
Dual Birth-Death Processes,and G.. Now let {λ.,μ.} ∈ G and {λ., μ.} = f({λ.,μ.})∈ G. be two related sets of birth-death parameters and {π.}, respectively {π.}, the associated potential coefficients. The following identities are easily verified in view of (1.3.3) and (3.1.2). .and(3.1.4)
形容词
发表于 2025-3-23 19:11:50
Preliminaries,., for every n ≥ 2, 0 ≤ t. <.....< t. and any i.,...., i. in S one has ., The process is supposed to be ., i.e., for every i, j in S the conditional probability Pr{X(t+s) = j| X(s) = i} does not depend on s. In this case we may put . t ≥ 0.
ELUDE
发表于 2025-3-23 22:30:25
Natural Birth-Death Processes,is and the following chapters we shall be concerned with natural birth-death processes only, i.e., A is assumed to satisfy the conditions C(A) and D(A). The state -1 will be disregarded and the term transition matrix will be used for the matrix P(⋅) = (p.(⋅)), where i, j = 0, 1,....... Since the pro
white-matter
发表于 2025-3-24 03:11:01
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Haphazard
发表于 2025-3-24 08:33:04
Stochastic Monotonicity: General Results,r. The initial distribution vector of {X(t)} will be denoted by . = (q., q.,....)., i.e.,.otherwise the notation of section 1.4 will be used. We have.where vector inequality is defined by (1.2.15) and . and . are the column vectors consisting of 0’s and 1’s, respectively. We recall that.where .(t) =
dithiolethione
发表于 2025-3-24 11:33:22
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euphoria
发表于 2025-3-24 18:10:05
The Truncated Birth-Death Process, with transition probability functions . which satisfy the conditions . and for i ∈ S = {0, 1,..., N},. as t → 0, where λ. and µ., i ∈ S, are non-negative constants. Throughout this chapter we assume λ. > 0 for i ∈ S{N} and µ. > 0 for i ∈ S{0}.
dilute
发表于 2025-3-24 22:28:03
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Exclaim
发表于 2025-3-24 23:31:33
0930-0325if the probabilities Pr{X(t) > i}, i E S, are increasing (decreasing) with t on T. Stochastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for