content_image/303153.jpg content_image/303152.jpg content_image/303151.jpg content_image/303148.jpg content_image/303150.jpg content_image/303149.jpg
## A Proof of Remark 3.1 

Suppose Assumption 2.2 is satisfied, lim$_{t}$$_{→}$$_{1}$ Ξ( t ) = ∞ and conditions (3.3) and (3.4) hold. Then L’Hˆ opital rule will give (notice that due to (3.4) and continuity of σ$_{z}$ ( t ) in the vicinity of 1 we have lim$_{t}$$_{→}$$_{1}$ ( 1 + σ 2 $_{z}$( t ) < 2) 

<img src='content_image/303153.jpg'>

Since by L’Hˆ opital rule we have 

it follows that 

<img src='content_image/303152.jpg'>

<img src='content_image/303151.jpg'>

where f ( t ) = ∫ t 0 1 Σ$_{z}$( s )+ σ $^{2}$− $_{s}$ds and lim$_{t}$$_{→}$$_{1}$ f ( t ) = ∞ . Since lim$_{x}$$_{→∞}$ log( x ) x α = 0, for any α > 0 we need to show that 

for some α > 0 to establish (3.6). 

Consider any α ∈ (0 , 1) and denote by 

then for t ≥ t ∗ we have 

<img src='content_image/303148.jpg'>

<img src='content_image/303150.jpg'>

<img src='content_image/303149.jpg'>

where c is some positive constant. Due to this expression and since lim$_{t}$$_{→}$$_{1}$ f ( t ) = ∞ , α < 1 and, 

