"Technical Analysis in Financial Markets"

equal for all agents, V(W_j,t^h|I_t)=σ_h,t² for all j. As a simple example we can take the return of the risky asset to be equal to the risk free rate for t=1,...,T. Then

E(W_j,T^h|I_T)=q_T^h R^T ω₀,

V(W_j,T^h|I_T)=q_T^h (1-q_T^h) (R^T ω₀)².

Hence in this simple example expected wealth and variance of wealth transferred by agent j to belief h both increase in time.

Fraction of total market wealth assigned to belief h by all agents

We define

W_j,t^h=

W_j,t^h

W_t

as the individual wealth assigned by agent j to belief h as a fraction of total market wealth W_t. The choices agents make at time t are dependent on the performances of the different beliefs until and including time t-1, hence the choices are independent of the price and wealth at time t. However, the price set at time t, which influences the wealth of each agent at time t and thus total wealth, is dependent on the choice of each agent at time t. Thus W_1,t^h, ... , W_N,t^h given I_t are dependent. However, if an agent is very small relative to the market, his choice will have a negligible effect on the eventual price set at time t. Hence if we assume that the market power of each agent is zero, that is

∀ t ∧ ∀ j:

lim

N → ∞

W_j,t

W_t

→ 0, (58)

then the law of large numbers still holds. Thus W_1,t^h, ... , W_N,t^h given I_t are dependent but identically distributed with mean E(W_j,t^h|I_t)=q_t^h ω_t/W_t and finite (under assumption 6.58) variance V(W_j,t^h|I_t), so that

∑

j=1

( W_j,t^h)=

W_t^h

W_t

→

q_t^h

ω_t

W_t

. (59)

This means that the average wealth per agent which is assigned to belief h as a fraction of total market wealth converges in probability to q_t^h ω_t/W_t as the number of agents goes to infinity. Average wealth per agent as a fraction of total wealth converges to:

W_t

∑

h=1

W_t^h

W_t

→

∑

h=1

q_t^h

ω_t

W_t

ω_t

W_t

. (60)

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